/*============================================================================
 * Manage the definition/setting of advection fields
 *============================================================================*/

/*
  This file is part of Code_Saturne, a general-purpose CFD tool.

  Copyright (C) 1998-2021 EDF S.A.

  This program is free software; you can redistribute it and/or modify it under
  the terms of the GNU General Public License as published by the Free Software
  Foundation; either version 2 of the License, or (at your option) any later
  version.

  This program is distributed in the hope that it will be useful, but WITHOUT
  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
  FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
  details.

  You should have received a copy of the GNU General Public License along with
  this program; if not, write to the Free Software Foundation, Inc., 51 Franklin
  Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/

/*----------------------------------------------------------------------------*/

#include "cs_defs.h"

/*----------------------------------------------------------------------------
 * Standard C library headers
 *----------------------------------------------------------------------------*/

#include <assert.h>
#include <ctype.h>
#include <float.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

/*----------------------------------------------------------------------------
 * Local headers
 *----------------------------------------------------------------------------*/

#include <bft_mem.h>

#include "cs_boundary_zone.h"
#include "cs_evaluate.h"
#include "cs_equation_common.h"
#include "cs_field.h"
#include "cs_log.h"
#include "cs_math.h"
#include "cs_mesh_location.h"
#include "cs_param_cdo.h"
#include "cs_reco.h"
#include "cs_volume_zone.h"
#include "cs_xdef.h"
#include "cs_zone.h"

/*----------------------------------------------------------------------------
 * Header for the current file
 *----------------------------------------------------------------------------*/

#include "cs_advection_field.h"

/*----------------------------------------------------------------------------*/

BEGIN_C_DECLS

/*=============================================================================
 * Local Macro definitions and structure definitions
 *============================================================================*/

/* Redefined names of function from cs_math to get shorter names */
#define _dp3 cs_math_3_dot_product

#define CS_ADVECTION_FIELD_DBG  0

#define CS_ADVECTION_FIELD_ID_NOT_SET      -1
#define CS_ADVECTION_FIELD_ID_TO_BE_SET    -2

/*============================================================================
 * Private variables
 *============================================================================*/

static const char _err_empty_adv[] =
  " Stop setting an empty cs_adv_field_t structure.\n"
  " Please check your settings.\n";

/* Pointer to shared structures (owned by a cs_domain_t structure) */
static const cs_cdo_quantities_t  *cs_cdo_quant;
static const cs_cdo_connect_t  *cs_cdo_connect;

  /* Advection fields attached to the computational domain */
static int  _n_adv_fields = 0;
static cs_adv_field_t  **_adv_fields = NULL;

/*============================================================================
 * Inline private function prototypes
 *============================================================================*/

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Return the required dimension for the definition of an advection
 *         field
 *
 * \param[in]      adv      pointer to an advection field structure
 *
 * \return the dimension
 */
/*----------------------------------------------------------------------------*/

static inline int
_get_dim_def(const cs_adv_field_t   *adv)
{
  int  dim = -1;

  if (adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR)
    dim = 3;
  else if (adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX)
    dim = 1;
  else
    bft_error(__FILE__, __LINE__, 0,
              "%s: Invalid dimension for the advection field.", __func__);

  return dim;
}

/*============================================================================
 * Private function prototypes
 *============================================================================*/

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Update the contribution of the flux
 *
 * \param[in]      cdoq       pointer to a cs_cdo_quantities_t structure
 * \param[in]      f2e        face --> edge connectivity
 * \param[in]      e2v        edge --> vertices connectivity
 * \param[in]      bf_id      boundary face id
 * \param[in]      face_flux  value of the normal flux across the face
 * \param[in, out] fluxes     array of values to update
 */
/*----------------------------------------------------------------------------*/

static void
_fill_uniform_boundary_flux(const cs_cdo_quantities_t  *const cdoq,
                            const cs_adjacency_t       *const f2e,
                            const cs_adjacency_t       *const e2v,
                            cs_lnum_t                   bf_id,
                            cs_real_t                   face_flux,
                            cs_real_t                  *fluxes)
{
  const cs_real_t  face_coef = 0.5*face_flux/cdoq->b_face_surf[bf_id];
  const cs_real_t  *xf = cdoq->b_face_center + 3*bf_id;
  const cs_lnum_t  f_id = cdoq->n_i_faces + bf_id;

  for (cs_lnum_t i = f2e->idx[f_id]; i < f2e->idx[f_id+1]; i++) {

    const cs_lnum_t  eshift = 2*f2e->ids[i];
    const cs_lnum_t  v0 = e2v->ids[eshift];
    const cs_lnum_t  v1 = e2v->ids[eshift+1];
    const double  tef = cs_math_surftri(cdoq->vtx_coord + 3*v0,
                                        cdoq->vtx_coord + 3*v1,
                                        xf);
    const double  weighted_flux = tef * face_coef;

    fluxes[v0] += weighted_flux;
    fluxes[v1] += weighted_flux;

  } /* Loop on face edges */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Update the contribution of the flux for each vertex
 *
 * \param[in]      cm         pointer to a cs_cell_mesh_t structure
 * \param[in]      f          face id in the cellwise numbering
 * \param[in]      face_flux  value of the normal flux across the face
 * \param[in, out] fluxes     normal boundary flux for each vertex of the face
 */
/*----------------------------------------------------------------------------*/

static void
_cw_fill_uniform_boundary_flux(const cs_cell_mesh_t   *cm,
                               short int               f,
                               cs_real_t               face_flux,
                               cs_real_t              *fluxes)
{
  const cs_real_t  face_coef = 0.5*face_flux/cm->face[f].meas;

  for (short int i = cm->f2e_idx[f]; i < cm->f2e_idx[f+1]; i++) {

    const short int  eshift = 2*cm->f2e_ids[i];
    const short int  v0 = cm->e2v_ids[eshift];
    const short int  v1 = cm->e2v_ids[eshift+1];
    const double  weighted_flux = face_coef * cm->tef[i];

    fluxes[v0] += weighted_flux;
    fluxes[v1] += weighted_flux;

  } /* Loop on face edges */
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute a vector-valued reconstruction of the advection field at
 *         vertices from the definition of the advection field
 *
 * \param[in]      quant        additional mesh quantities struct.
 * \param[in]      connect      pointer to a cs_cdo_connect_t struct.
 * \param[in]      def          pointer to a cs_xdef_t struct.
 * \param[in, out] vtx_values   values of the reconstruction
 */
/*----------------------------------------------------------------------------*/

static void
_compute_adv_vector_at_vertices(const cs_cdo_quantities_t  *quant,
                                const cs_cdo_connect_t     *connect,
                                cs_xdef_t                  *def,
                                cs_real_t                  *vtx_values)
{
  memset(vtx_values, 0, 3*quant->n_vertices*sizeof(cs_real_t));

  const cs_adjacency_t  *c2v = connect->c2v;

  switch (def->type) {

  case CS_XDEF_BY_ARRAY:
    {
      cs_real_t  cell_vector[3];
      cs_xdef_array_context_t  *ai = (cs_xdef_array_context_t *)def->context;

      if (cs_flag_test(ai->loc, cs_flag_dual_face_byc)) {

        assert(ai->index == connect->c2e->idx);

        for (cs_lnum_t c_id = 0; c_id < quant->n_cells; c_id++) {

          cs_reco_dfbyc_at_cell_center(c_id,
                                       connect->c2e,
                                       quant,
                                       ai->values,
                                       cell_vector);

          for (cs_lnum_t j = c2v->idx[c_id]; j < c2v->idx[c_id+1]; j++) {

            const cs_real_t  vc_vol = quant->dcell_vol[j];
            cs_real_t  *_val = vtx_values + 3*c2v->ids[j];

            _val[0] += vc_vol * cell_vector[0];
            _val[1] += vc_vol * cell_vector[1];
            _val[2] += vc_vol * cell_vector[2];

          }

        } /* Loop on cells */

      }
      else if (cs_flag_test(ai->loc, cs_flag_primal_face)) {

        for (cs_lnum_t c_id = 0; c_id < quant->n_cells; c_id++) {

          cs_reco_cell_vector_by_face_dofs(c_id,
                                           connect->c2f,
                                           quant,
                                           ai->values,
                                           false, /* local input ? */
                                           cell_vector);

          for (cs_lnum_t j = c2v->idx[c_id]; j < c2v->idx[c_id+1]; j++) {

            const cs_real_t  vc_vol = quant->dcell_vol[j];
            cs_real_t  *_val = vtx_values + 3*c2v->ids[j];

            _val[0] += vc_vol * cell_vector[0];
            _val[1] += vc_vol * cell_vector[1];
            _val[2] += vc_vol * cell_vector[2];

          }

        } /* Loop on cells */

      }
      else
        bft_error(__FILE__, __LINE__, 0,
                  " %s: Invalid location for array", __func__);

    }
    break; /* by array */

  case CS_XDEF_BY_DOF_FUNCTION:
    {
#if defined(HAVE_OPENMP) /* Determine the default number of OpenMP threads */
      int  t_id = omp_get_thread_num();
#else
      int  t_id = 0;
#endif
      cs_real_t  cell_vector[3];
      cs_real_t  *fluxes = cs_cdo_local_get_d_buffer(t_id);
      cs_xdef_dof_context_t  *cx = (cs_xdef_dof_context_t *)def->context;

      if (cs_flag_test(cx->loc, cs_flag_primal_face) == false)
        bft_error(__FILE__, __LINE__, 0,
                  "%s: Invalid location for definition by DoFs.\n", __func__);

      const cs_lnum_t  *idx = connect->c2f->idx;

      for (cs_lnum_t c_id = 0; c_id < quant->n_cells; c_id++) {

        cs_lnum_t  n_fc = idx[c_id+1] - idx[c_id];
        const cs_lnum_t  *_ids = connect->c2f->ids + idx[c_id];

        /* Values of the function are defined at the primal faces */
        cx->func(n_fc, _ids, true, /* dense output */
                 cx->input,
                 fluxes);

        cs_reco_cell_vector_by_face_dofs(c_id,
                                         connect->c2f,
                                         quant,
                                         fluxes,
                                         true, /* local input ? */
                                         cell_vector);

        for (cs_lnum_t j = c2v->idx[c_id]; j < c2v->idx[c_id+1]; j++) {

          const cs_real_t  vc_vol = quant->dcell_vol[j];
          cs_real_t  *_val = vtx_values + 3*c2v->ids[j];

          _val[0] += vc_vol * cell_vector[0];
          _val[1] += vc_vol * cell_vector[1];
          _val[2] += vc_vol * cell_vector[2];

        }

      } /* Loop on cells */

    }
    break;

  default:
    bft_error(__FILE__, __LINE__, 0, " %s: Incompatible type of definition.",
              __func__);
    break;

  } /* Type of definition */

}

/*============================================================================
 * Public function prototypes
 *============================================================================*/

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Set shared pointers to main domain members
 *
 * \param[in]  quant       additional mesh quantities struct.
 * \param[in]  connect     pointer to a cs_cdo_connect_t struct.
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_set_shared_pointers(const cs_cdo_quantities_t  *quant,
                                       const cs_cdo_connect_t     *connect)
{
  /* Assign static const pointers */
  cs_cdo_quant = quant;
  cs_cdo_connect = connect;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Get the number of allocated cs_adv_field_t structures
 *
 * \return the number of advection fields
 */
/*----------------------------------------------------------------------------*/

int
cs_advection_field_get_n_fields(void)
{
  return _n_adv_fields;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Search in the array of advection field structures which one has
 *         the name given in argument
 *
 * \param[in]  name        name of the advection field
 *
 * \return a pointer to a cs_adv_field_t structure or NULL if not found
 */
/*----------------------------------------------------------------------------*/

cs_adv_field_t *
cs_advection_field_by_name(const char   *name)
{
  if (_n_adv_fields <= 0)
    return NULL;

  for (int i = 0; i < _n_adv_fields; i++) {

    cs_adv_field_t  *adv = _adv_fields[i];
    if (cs_advection_field_check_name(adv, name))
      return adv;

  }

  return NULL;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Search in the array of advection field structures which one has
 *         the id given in argument
 *
 * \param[in]  id        identification number
 *
 * \return a pointer to a cs_adv_field_t structure or NULL if not found
 */
/*----------------------------------------------------------------------------*/

cs_adv_field_t *
cs_advection_field_by_id(int      id)
{
  if (_n_adv_fields <= 0)
    return NULL;
  if (id >= _n_adv_fields || id < 0)
    return NULL;
  if (_adv_fields == NULL)
    return NULL;

  return _adv_fields[id];
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Add and initialize a new user-defined advection field structure
 *
 * \param[in]  name        name of the advection field
 *
 * \return a pointer to the new allocated cs_adv_field_t structure
 */
/*----------------------------------------------------------------------------*/

cs_adv_field_t *
cs_advection_field_add_user(const char  *name)
{
  return cs_advection_field_add(name, CS_ADVECTION_FIELD_USER);
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Add and initialize a new advection field structure
 *
 * \param[in]  name        name of the advection field
 * \param[in]  status      status of the advection field to create
 *
 * \return a pointer to the new allocated cs_adv_field_t structure
 */
/*----------------------------------------------------------------------------*/

cs_adv_field_t *
cs_advection_field_add(const char                    *name,
                       cs_advection_field_status_t    status)
{
  if (name == NULL)
    bft_error(__FILE__, __LINE__, 0,
              "%s: A non-empty name is mandatory to add a new advection field",
              __func__);

  cs_adv_field_t  *adv = cs_advection_field_by_name(name);
  if (adv != NULL) {
    cs_base_warn(__FILE__, __LINE__);
    cs_log_printf(CS_LOG_DEFAULT,
                  _(" An existing advection field has already the name %s.\n"
                    " Stop adding this advection field.\n"), name);
    return adv;
  }

  /* Sanity check */
  if (!(status & CS_ADVECTION_FIELD_USER) &&
      !(status & CS_ADVECTION_FIELD_GWF)  &&
      !(status & CS_ADVECTION_FIELD_NAVSTO))
    bft_error(__FILE__, __LINE__, 0,
              "%s: No category associated to the advection field %s.",
              __func__, name);

  int  new_id = _n_adv_fields;
  _n_adv_fields++;
  BFT_REALLOC(_adv_fields, _n_adv_fields, cs_adv_field_t *);
  _adv_fields[new_id] = NULL;

  BFT_MALLOC(adv, 1, cs_adv_field_t);

  /* Copy name */
  int  len = strlen(name) + 1;
  BFT_MALLOC(adv->name, len, char);
  strncpy(adv->name, name, len);

  adv->id = new_id;
  adv->status = status;
  adv->post_flag = 0;

  /* Default initialization */
  adv->definition = NULL;

  adv->n_bdy_flux_defs = 0;
  adv->bdy_flux_defs = NULL;
  adv->bdy_def_ids = NULL;

  adv->cell_field_id = CS_ADVECTION_FIELD_ID_NOT_SET;
  adv->int_field_id = CS_ADVECTION_FIELD_ID_NOT_SET;

  adv->vtx_field_id = CS_ADVECTION_FIELD_ID_NOT_SET;
  if (adv->status & CS_ADVECTION_FIELD_DEFINE_AT_VERTICES)
    adv->vtx_field_id = CS_ADVECTION_FIELD_ID_TO_BE_SET;

  adv->bdy_field_id = CS_ADVECTION_FIELD_ID_NOT_SET;
  if (adv->status & CS_ADVECTION_FIELD_DEFINE_AT_BOUNDARY_FACES)
    adv->bdy_field_id = CS_ADVECTION_FIELD_ID_TO_BE_SET;

  /* Update the status to a scalar flux in case of an advection field coming
     from the legacy FV part */
  if ((status & CS_ADVECTION_FIELD_NAVSTO) &&
      (status & CS_ADVECTION_FIELD_LEGACY_FV))
    adv->status |= CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX;

  /* If no type is set, set vector-valued by default */
  if (!(status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX) &&
      !(status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR))
    adv->status |= CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR;

  /* Store the new advection field */
  _adv_fields[new_id] = adv;

  return adv;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Free all allocated cs_adv_field_t structures and its related array
 *
 * \return a NULL pointer
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_destroy_all(void)
{
  if (_adv_fields == NULL)
    return;

  for (int i = 0; i < _n_adv_fields; i++) {

    cs_adv_field_t  *adv = _adv_fields[i];

    adv->definition = cs_xdef_free(adv->definition);

    for (int id = 0; id < adv->n_bdy_flux_defs; id++)
      adv->bdy_flux_defs[id] = cs_xdef_free(adv->bdy_flux_defs[id]);

    if (adv->n_bdy_flux_defs > 0) BFT_FREE(adv->bdy_flux_defs);
    if (adv->bdy_def_ids != NULL)   BFT_FREE(adv->bdy_def_ids);

    BFT_FREE(adv->name);
    BFT_FREE(adv);

    /* All other pointers are shared */

  }  /* Loop on advection fields */

  BFT_FREE(_adv_fields);
  _n_adv_fields = 0;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Check if the given advection field has the name ref_name
 *
 * \param[in]  adv         pointer to a cs_adv_field_t structure to test
 * \param[in]  ref_name    name of the advection field to find
 *
 * \return true if the name of the advection field is ref_name otherwise false
 */
/*----------------------------------------------------------------------------*/

bool
cs_advection_field_check_name(const cs_adv_field_t   *adv,
                              const char             *ref_name)
{
  if (adv == NULL)
    return false;

  int  reflen = strlen(ref_name);
  int  len = strlen(adv->name);

  if (reflen == len) {
    if (strcmp(ref_name, adv->name) == 0)
      return true;
    else
      return false;
  }
  else
    return false;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Print all setup information related to cs_adv_field_t structures
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_log_setup(void)
{
  if (_adv_fields == NULL)
    return;

  char  prefix[256];

  cs_log_printf(CS_LOG_SETUP, "\nSummary of the advection field\n");
  cs_log_printf(CS_LOG_SETUP, "%s\n", cs_sep_h1);

  for (int i = 0; i < _n_adv_fields; i++) {

    const cs_adv_field_t  *adv = _adv_fields[i];

    if (adv == NULL)
      continue;
    assert(strlen(adv->name) < 200);

    /* Category of advection field */
    cs_log_printf(CS_LOG_SETUP, "  * %s | Category: ", adv->name);
    if (adv->status &  CS_ADVECTION_FIELD_NAVSTO)
      cs_log_printf(CS_LOG_SETUP, "Related to Navier-Stokes\n");
    else if (adv->status & CS_ADVECTION_FIELD_GWF)
      cs_log_printf(CS_LOG_SETUP,
                    "Related to the \"Groundwater Flow\" module\n");
    else if (CS_ADVECTION_FIELD_USER)
      cs_log_printf(CS_LOG_SETUP, "User-defined\n");

    /* Type of advection field */
    cs_log_printf(CS_LOG_SETUP, "  * %s | Type: ", adv->name);
    if (adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR)
      cs_log_printf(CS_LOG_SETUP, "Velocity vector\n");
    else if (adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX)
      cs_log_printf(CS_LOG_SETUP, "Scalar flux\n");

    /* Additional information */
    if (adv->status & CS_ADVECTION_FIELD_LEGACY_FV)
      cs_log_printf(CS_LOG_SETUP, "  * %s | %s\n",
                    adv->name, "Related to Legacy FV schemes\n");
    if (adv->status & CS_ADVECTION_FIELD_STEADY)
      cs_log_printf(CS_LOG_SETUP,
                    "  * %s | Time status: Steady-state\n", adv->name);
    else
      cs_log_printf(CS_LOG_SETUP,
                    "  * %s | Time status: Unsteady\n", adv->name);

    if (adv->post_flag & CS_ADVECTION_FIELD_POST_COURANT)
      cs_log_printf(CS_LOG_SETUP,
                    "  * %s | Postprocess the Courant number\n", adv->name);

    /* Where fields are defined */
    bool  at_cells =
      (adv->cell_field_id > CS_ADVECTION_FIELD_ID_NOT_SET) ? true : false;
    bool  at_vertices =
      (adv->vtx_field_id > CS_ADVECTION_FIELD_ID_NOT_SET) ? true : false;
    bool  at_bfaces =
      (adv->bdy_field_id > CS_ADVECTION_FIELD_ID_NOT_SET) ? true : false;
    bool  at_ifaces =
      (adv->int_field_id > CS_ADVECTION_FIELD_ID_NOT_SET) ? true : false;

    cs_log_printf(CS_LOG_SETUP,
                  "  * %s | Fields defined at cells: %s; at vertices: %s\n",
                  adv->name, cs_base_strtf(at_cells),
                  cs_base_strtf(at_vertices));
    cs_log_printf(CS_LOG_SETUP,
                  "  * %s | Fields defined at boundary faces: %s;"
                  " at interior faces: %s\n\n",
                  adv->name, cs_base_strtf(at_bfaces),
                  cs_base_strtf(at_ifaces));

    sprintf(prefix, "        Definition");
    cs_xdef_log(prefix, adv->definition);

    /* Boundary flux definition */
    cs_log_printf(CS_LOG_SETUP,
                  "  * %s | Number of boundary flux definitions: %d\n",
                  adv->name, adv->n_bdy_flux_defs);
    if (adv->n_bdy_flux_defs > 0)
      cs_log_printf(CS_LOG_SETUP, "\n");
    for (int ib = 0; ib < adv->n_bdy_flux_defs; ib++) {
      sprintf(prefix, "        Definition %2d", ib);
      cs_xdef_log(prefix, adv->bdy_flux_defs[ib]);
    }

  }  /* Loop on advection fields */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Set optional post-processings
 *
 * \param[in, out]  adv         pointer to a cs_adv_field_t structure
 * \param[in]       post_flag   flag to set
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_set_postprocess(cs_adv_field_t            *adv,
                                   cs_flag_t                  post_flag)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  adv->post_flag |= post_flag;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define the value of a cs_adv_field_t structure
 *
 * \param[in, out]  adv       pointer to a cs_adv_field_t structure
 * \param[in]       values    values to set
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_by_value(cs_adv_field_t    *adv,
                                cs_real_t         *values)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  cs_flag_t  state_flag = CS_FLAG_STATE_UNIFORM |
    CS_FLAG_STATE_CELLWISE | CS_FLAG_STATE_STEADY;
  cs_flag_t  meta_flag = CS_FLAG_FULL_LOC;
  int  dim = _get_dim_def(adv);

  adv->definition = cs_xdef_volume_create(CS_XDEF_BY_VALUE,
                                          dim,
                                          0,  /* zone_id = 0 => all cells */
                                          state_flag,
                                          meta_flag,
                                          values);
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define a cs_adv_field_t structure thanks to an analytic function
 *
 * \param[in, out]  adv     pointer to a cs_adv_field_t structure
 * \param[in]       func    pointer to a function
 * \param[in]       input   NULL or pointer to a structure cast on-the-fly
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_by_analytic(cs_adv_field_t        *adv,
                                   cs_analytic_func_t    *func,
                                   void                  *input)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  cs_flag_t  state_flag = 0;
  cs_flag_t  meta_flag = CS_FLAG_FULL_LOC;
  int  dim = _get_dim_def(adv);
  cs_xdef_analytic_context_t  ac = { .z_id = 0, /* all cells */
                                     .func = func,
                                     .input = input,
                                     .free_input = NULL };

  adv->definition = cs_xdef_volume_create(CS_XDEF_BY_ANALYTIC_FUNCTION,
                                          dim,
                                          0,  /* zone_id = 0 => all cells */
                                          state_flag,
                                          meta_flag,
                                          &ac);
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define a cs_adv_field_t structure using a cs_dof_func_t
 *
 * \param[in, out] adv       pointer to a cs_adv_field_t structure
 * \param[in]      loc_flag  location where values are computed
 * \param[in]      func      pointer to a cs_dof_func_t function
 * \param[in]      input     NULL or pointer to a structure cast on-the-fly
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_by_dof_func(cs_adv_field_t    *adv,
                                   cs_flag_t          loc_flag,
                                   cs_dof_func_t     *func,
                                   void              *input)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  cs_flag_t  state_flag = 0;
  cs_flag_t  meta_flag = CS_FLAG_FULL_LOC;
  int  dim = _get_dim_def(adv);
  cs_xdef_dof_context_t  cx = { .z_id = 0, /* all cells */
                                .loc = loc_flag,
                                .func = func,
                                .input = input,
                                .free_input = NULL };

  adv->definition = cs_xdef_volume_create(CS_XDEF_BY_DOF_FUNCTION,
                                          dim,
                                          0,  /* zone_id = 0 => all cells */
                                          state_flag,
                                          meta_flag,
                                          &cx);
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define a cs_adv_field_t structure thanks to an array of values
 *
 * \param[in, out]  adv       pointer to a cs_adv_field_t structure
 * \param[in]       loc       information to know where are located values
 * \param[in]       array     pointer to an array
 * \param[in]       is_owner  transfer the lifecycle to the cs_xdef_t structure
 *                            (true or false)
 * \param[in]       index     optional pointer to the array index
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_by_array(cs_adv_field_t    *adv,
                                cs_flag_t          loc,
                                cs_real_t         *array,
                                bool               is_owner,
                                cs_lnum_t         *index)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  cs_flag_t  state_flag = 0; /* Will be updated during the creation */
  cs_flag_t  meta_flag = CS_FLAG_FULL_LOC;
  cs_xdef_array_context_t  context = {.z_id = 0, /* all cells */
                                      .stride = 3, /* default initialization */
                                      .loc = loc,
                                      .values = array,
                                      .is_owner = is_owner,
                                      .index = index };

  /* Set the stride accord to the status (flux or vector) */
  context.stride = _get_dim_def(adv);

  /* Checkings */
  if ((loc & CS_FLAG_SCALAR) && context.stride == 3)
    bft_error(__FILE__, __LINE__, 0,
              "%s: Incompatible setting for advection field %s\n"
              " Array is set as a flux while the advection field as a vector.",
              __func__, adv->name);

  if ((loc & CS_FLAG_VECTOR) && context.stride == 1)
    bft_error(__FILE__, __LINE__, 0,
              "%s: Incompatible setting for advection field %s\n"
              " Array is set as a vector while the advection field as a flux.",
              __func__, adv->name);

  adv->definition = cs_xdef_volume_create(CS_XDEF_BY_ARRAY,
                                          context.stride,
                                          0,  /* zone_id = all cells */
                                          state_flag,
                                          meta_flag,
                                          (void *)&context);
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define a cs_adv_field_t structure thanks to a field structure
 *
 * \param[in, out]  adv       pointer to a cs_adv_field_t structure
 * \param[in]       field     pointer to a cs_field_t structure
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_by_field(cs_adv_field_t    *adv,
                                cs_field_t        *field)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  /* Flags will be updated during the creation */
  cs_flag_t  state_flag = 0;
  cs_flag_t  meta_flag = 0;

  /* Set the stride accord to the status (flux or vector) */
  int  dim = _get_dim_def(adv);
  if (field->dim != dim)
    bft_error(__FILE__, __LINE__, 0,
              " %s: Inconsistency found between the field dimension and the"
              " definition of the advection field %s.\n",
              __func__, adv->name);

  adv->definition = cs_xdef_volume_create(CS_XDEF_BY_FIELD,
                                          dim,
                                          0,  /* zone_id */
                                          state_flag,
                                          meta_flag,
                                          field);
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define the value of the boundary normal flux for the given
 *         cs_adv_field_t structure
 *
 * \param[in, out]  adv           pointer to a cs_adv_field_t structure
 * \param[in]       zname         name of the boundary zone to consider
 * \param[in]       normal_flux   value to set
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_boundary_flux_by_value(cs_adv_field_t    *adv,
                                              const char        *zname,
                                              cs_real_t          normal_flux)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  cs_flag_t  state_flag = CS_FLAG_STATE_UNIFORM | CS_FLAG_STATE_FACEWISE;
  cs_flag_t  meta_flag = 0;

  cs_xdef_t  *d = cs_xdef_boundary_create(CS_XDEF_BY_VALUE,
                                          1,  /* dim. */
                                          cs_get_bdy_zone_id(zname),
                                          state_flag,
                                          meta_flag,
                                          (void *)&normal_flux);

  int  def_id = adv->n_bdy_flux_defs;
  adv->n_bdy_flux_defs += 1;
  BFT_REALLOC(adv->bdy_flux_defs, adv->n_bdy_flux_defs, cs_xdef_t *);
  adv->bdy_flux_defs[def_id] = d;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define the value of the boundary normal flux for the given
 *         cs_adv_field_t structure using an analytic function
 *
 * \param[in, out]  adv     pointer to a cs_adv_field_t structure
 * \param[in]       zname   name of the boundary zone to consider
 * \param[in]       func    pointer to a function
 * \param[in]       input   NULL or pointer to a structure cast on-the-fly
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_boundary_flux_by_analytic(cs_adv_field_t        *adv,
                                                 const char            *zname,
                                                 cs_analytic_func_t    *func,
                                                 void                  *input)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  cs_flag_t  state_flag = 0, meta_flag = 0;
  int  z_id = cs_get_bdy_zone_id(zname);
  cs_xdef_analytic_context_t  ac = { .z_id = z_id,
                                     .func = func,
                                     .input = input,
                                     .free_input = NULL };

  cs_xdef_t  *d = cs_xdef_boundary_create(CS_XDEF_BY_ANALYTIC_FUNCTION,
                                          1,  /* dim. */
                                          z_id,
                                          state_flag,
                                          meta_flag,
                                          &ac);

  int  def_id = adv->n_bdy_flux_defs;
  adv->n_bdy_flux_defs += 1;
  BFT_REALLOC(adv->bdy_flux_defs, adv->n_bdy_flux_defs, cs_xdef_t *);
  adv->bdy_flux_defs[def_id] = d;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define the value of the boundary normal flux for the given
 *         cs_adv_field_t structure using an array of values
 *
 * \param[in, out]  adv       pointer to a cs_adv_field_t structure
 * \param[in]       zname     name of the boundary zone to consider
 * \param[in]       loc       information to know where are located values
 * \param[in]       array     pointer to an array
 * \param[in]       is_owner  transfer the lifecycle to the cs_xdef_t structure
 *                            (true or false)
 * \param[in]       index     optional pointer to the array index
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_boundary_flux_by_array(cs_adv_field_t    *adv,
                                              const char        *zname,
                                              cs_flag_t          loc,
                                              cs_real_t         *array,
                                              bool               is_owner,
                                              cs_lnum_t         *index)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  if (loc & CS_FLAG_VECTOR)
    bft_error(__FILE__, __LINE__, 0,
              "%s: Advection field: %s\n"
              " The boundary flux is not compatible with a vector-valued"
              " definition.\n", __func__, adv->name);

  cs_flag_t  state_flag =  0;
  cs_flag_t  meta_flag = 0;

  int  z_id = cs_get_bdy_zone_id(zname);
  if (z_id == 0)
    meta_flag  |= CS_FLAG_FULL_LOC;

  cs_xdef_array_context_t  context = {.z_id = z_id,
                                      .stride = 1,
                                      .loc = loc,
                                      .values = array,
                                      .is_owner = is_owner,
                                      .index = index };

  cs_xdef_t  *d = cs_xdef_boundary_create(CS_XDEF_BY_ARRAY,
                                          1,  /* dim. */
                                          z_id,
                                          state_flag,
                                          meta_flag,
                                          (void *)&context);

  int  def_id = adv->n_bdy_flux_defs;
  adv->n_bdy_flux_defs += 1;
  BFT_REALLOC(adv->bdy_flux_defs, adv->n_bdy_flux_defs, cs_xdef_t *);
  adv->bdy_flux_defs[def_id] = d;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Define the value of the boundary normal flux for the given
 *         cs_adv_field_t structure using a field structure
 *
 * \param[in, out]  adv       pointer to a cs_adv_field_t structure
 * \param[in]       field     pointer to a cs_field_t structure
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_def_boundary_flux_by_field(cs_adv_field_t    *adv,
                                              cs_field_t        *field)
{
  if (adv == NULL)
    bft_error(__FILE__, __LINE__, 0, _(_err_empty_adv));

  /* Flags will be updated during the creation */
  cs_flag_t  state_flag = 0;
  cs_flag_t  meta_flag = CS_FLAG_FULL_LOC; /* all boundary faces are handled */

  /* Set the stride accord to the status (flux or vector) */
  if (field->dim != 1)
    bft_error(__FILE__, __LINE__, 0,
              " %s: Inconsistency found in the field dimension.\n"
              " A flux is requested (dim = 1) for advection field %s\n",
              __func__, adv->name);

  cs_xdef_t  *d = cs_xdef_boundary_create(CS_XDEF_BY_FIELD,
                                          1,  /* dim. */
                                          0,  /* all boundary faces */
                                          state_flag,
                                          meta_flag,
                                          field);

  int  def_id = adv->n_bdy_flux_defs;
  assert(def_id == 0);
  adv->n_bdy_flux_defs += 1;
  BFT_REALLOC(adv->bdy_flux_defs, adv->n_bdy_flux_defs, cs_xdef_t *);
  adv->bdy_flux_defs[def_id] = d;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Create all needed cs_field_t structures related to an advection
 *         field
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_create_fields(void)
{
  int  len;
  char *field_name = NULL;

  for (int i = 0; i < _n_adv_fields; i++) {

    cs_adv_field_t  *adv = _adv_fields[i];
    assert(adv != NULL);

    bool  has_previous = (adv->status & CS_ADVECTION_FIELD_STEADY) ? true:false;
    int  field_mask = CS_FIELD_PROPERTY | CS_FIELD_CDO;

    /* Always add a field attached to cells (it may be used to define the
       numerical flux for advection, to compute adimensional numbers or to
       postprocess the advection field in case of definition by flux */
    if (adv->cell_field_id < 0) {

      if (cs_flag_test(adv->status, CS_ADVECTION_FIELD_NAVSTO)) {

        /* If this is the advection field related to the Navier-Stokes equations
           then there is no need to allocate a new buffer. One links to the
           existing velocity field */
        adv->cell_field_id = cs_field_id_by_name("velocity");
        assert(adv->cell_field_id > -1);

      }
      else {

        /* Define the name of the new field related the cell array */
        len = strlen(adv->name) + strlen("_cells") + 1;
        BFT_MALLOC(field_name, len, char);
        sprintf(field_name, "%s_cells", adv->name);

        cs_field_t  *fld = cs_field_create(field_name,
                                           field_mask,
                                           CS_MESH_LOCATION_CELLS,
                                           3, /* always a vector-valued field */
                                           has_previous);

        cs_field_set_key_int(fld, cs_field_key_id("log"), 1);
        cs_field_set_key_int(fld, cs_field_key_id("post_vis"), 1);

        adv->cell_field_id = cs_field_id_by_name(field_name);

        BFT_FREE(field_name);

      }

    } /* Add a field at cells */

    if (adv->vtx_field_id == CS_ADVECTION_FIELD_ID_TO_BE_SET) {

      /* The creation of a field to store the value at vertices has been
         requested. Add this field */
      len = strlen(adv->name) + strlen("_vertices") + 1;
      BFT_MALLOC(field_name, len, char);
      sprintf(field_name, "%s_vertices", adv->name);

      cs_field_t  *fld = cs_field_create(field_name,
                                         field_mask,
                                         CS_MESH_LOCATION_VERTICES,
                                         3,  /* always a vector-valued field */
                                         has_previous);

      cs_field_set_key_int(fld, cs_field_key_id("log"), 1);
      cs_field_set_key_int(fld, cs_field_key_id("post_vis"), 1);

      adv->vtx_field_id = cs_field_id_by_name(field_name);

      BFT_FREE(field_name);

    } /* Add a field attached to vertices */

    if (adv->bdy_field_id == CS_ADVECTION_FIELD_ID_TO_BE_SET) {

      /* Create a new field at the boundary faces for taking
         into account the normal flux used in the treatment of the boundary
         conditions for instance */
      len = strlen(adv->name) + strlen("_boundary_flux") + 1;
      BFT_MALLOC(field_name, len, char);
      sprintf(field_name, "%s_boundary_flux", adv->name);

      cs_field_t  *fld = cs_field_create(field_name,
                                         field_mask,
                                         CS_MESH_LOCATION_BOUNDARY_FACES,
                                         1,   /* always a scalar-valued field */
                                         has_previous);

      cs_field_set_key_int(fld, cs_field_key_id("log"), 1);
      cs_field_set_key_int(fld, cs_field_key_id("post_vis"), 1);

      adv->bdy_field_id = cs_field_id_by_name(field_name);

      BFT_FREE(field_name);

    } /* Add a field attached to boundary faces */

  } /* Loop on advection fields */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Last stage of the definition of an advection field based on several
 *         definitions (i.e. definition by subdomains on the boundary)
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_finalize_setup(void)
{
  if (_n_adv_fields == 0)
    return;

  for (int i = 0; i < _n_adv_fields; i++) {

    cs_adv_field_t  *adv = _adv_fields[i];
    assert(adv != NULL);

    /* Case of an advection field defined as the mass flux from the legacy FV
       part */
    if ((adv->status & CS_ADVECTION_FIELD_NAVSTO) &&
        (adv->status & CS_ADVECTION_FIELD_LEGACY_FV)) {

      /* Automatic definition and checkings */
      cs_field_t  *fld = cs_field_by_name("boundary_mass_flux");
      assert(fld != NULL);
      adv->bdy_field_id = fld->id;

      if (adv->bdy_flux_defs == NULL)
        cs_advection_field_def_boundary_flux_by_field(adv, fld);

      else { /* Sanity check */
        if (adv->n_bdy_flux_defs > 1 ||
            adv->bdy_flux_defs[0]->type != CS_XDEF_BY_FIELD)
          bft_error(__FILE__, __LINE__, 0,
                    "%s: Invalid setting found for the advection field %s\n"
                    " No need to perform additional setting when the legacy"
                    " FV mass flux is used.\n", __func__, adv->name);
      }

      fld = cs_field_by_name("inner_mass_flux");
      assert(fld != NULL);
      cs_advection_field_def_by_field(adv, fld);
      adv->int_field_id = fld->id;

      if (adv->definition == NULL)
        cs_advection_field_def_by_field(adv, fld);

      else { /* Sanity check */
        if (adv->definition->type != CS_XDEF_BY_FIELD)
          bft_error(__FILE__, __LINE__, 0,
                    "%s: Invalid setting found for the advection field %s\n"
                    " No need to perform additional setting when the legacy"
                    " FV mass flux is used.\n", __func__, adv->name);
      }

    }

    if (adv->n_bdy_flux_defs > 1) {

      const cs_lnum_t  n_b_faces = cs_cdo_quant->n_b_faces;

      BFT_MALLOC(adv->bdy_def_ids, n_b_faces, short int);
#     pragma omp parallel for if (n_b_faces > CS_THR_MIN)
      for (cs_lnum_t j = 0; j < n_b_faces; j++)
        adv->bdy_def_ids[j] = -1;  /* Unset by default */

      for (short int def_id = 0; def_id < adv->n_bdy_flux_defs; def_id++) {

        const cs_xdef_t  *def = adv->bdy_flux_defs[def_id];
        assert(def->z_id > 0);
        assert(def->support == CS_XDEF_SUPPORT_BOUNDARY);

        const cs_zone_t  *z = cs_boundary_zone_by_id(def->z_id);
        assert(z != NULL);

#       pragma omp parallel for if (z->n_elts > CS_THR_MIN)
        for (cs_lnum_t j = 0; j < z->n_elts; j++)
          adv->bdy_def_ids[z->elt_ids[j]] = def_id;

      }

    } /* More than one definition */

  } /* Loop on advection fields */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the value of the advection field at the cell center
 *
 * \param[in]      c_id    id of the current cell
 * \param[in]      adv     pointer to a cs_adv_field_t structure
 * \param[in, out] vect    pointer to a cs_nvec3_t structure (meas + unitv)
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_get_cell_vector(cs_lnum_t               c_id,
                                   const cs_adv_field_t   *adv,
                                   cs_nvec3_t             *vect)
{
  /* Initialize the vector */
  vect->meas = 0.;
  for (int k = 0; k < 3; k++)
    vect->unitv[k] = 0;

  if (adv == NULL)
    return;

  cs_field_t  *f = cs_advection_field_get_field(adv, CS_MESH_LOCATION_CELLS);

  cs_nvec3(f->val + 3*c_id, vect);
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the vector-valued interpolation of the advection field at
 *         a given location inside a cell
 *
 * \param[in]      adv          pointer to a cs_adv_field_t structure
 * \param[in]      cm           pointer to a cs_cell_mesh_t structure
 * \param[in]      xyz          location where to perform the evaluation
 * \param[in]      time_eval    physical time at which one evaluates the term
 * \param[in, out] eval         pointer to a cs_nvec3_t
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_cw_eval_at_xyz(const cs_adv_field_t  *adv,
                                  const cs_cell_mesh_t  *cm,
                                  const cs_real_3_t      xyz,
                                  cs_real_t              time_eval,
                                  cs_nvec3_t            *eval)
{
  if (adv == NULL)
    return;

  cs_xdef_t  *def = adv->definition;
  cs_real_3_t  vector_values = {0, 0, 0};

  if (adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX) {

    assert(def->dim == 1);
    switch (def->type) {

    case CS_XDEF_BY_ARRAY:        /* P0 approximation */
      {
        cs_xdef_array_context_t  *ai = (cs_xdef_array_context_t *)def->context;

        if (cs_flag_test(ai->loc, cs_flag_dual_face_byc)) {

          assert(ai->index == cs_cdo_connect->c2e->idx);
          cs_reco_dfbyc_in_cell(cm,
                                ai->values + ai->index[cm->c_id],
                                vector_values);

        }
        else if (cs_flag_test(ai->loc, cs_flag_primal_face)) {

          const cs_real_t  *i_flux = ai->values;
          const cs_real_t  *b_flux = ai->values + cs_cdo_quant->n_i_faces;

          cs_reco_cw_cell_vect_from_face_dofs(cm,
                                              i_flux,
                                              b_flux,
                                              vector_values);
        }
        else
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid location for array", __func__);

        cs_nvec3(vector_values, eval);
      }
      break;

    case CS_XDEF_BY_FIELD:          /* P0 approximation */
      {
        const cs_real_t  *i_flux = (cs_field_by_id(adv->int_field_id))->val;
        const cs_real_t  *b_flux = (cs_field_by_id(adv->bdy_field_id))->val;

        cs_reco_cw_cell_vect_from_face_dofs(cm,
                                            i_flux,
                                            b_flux,
                                            vector_values);
        cs_nvec3(vector_values, eval);
      }
      break;

    case CS_XDEF_BY_DOF_FUNCTION:   /* P0 approximation */
      {
#if defined(HAVE_OPENMP) /* Determine the default number of OpenMP threads */
        int  t_id = omp_get_thread_num();
#else
        int  t_id = 0;
#endif
        cs_real_t  *fluxes = cs_cdo_local_get_d_buffer(t_id);
        cs_xdef_dof_context_t  *cx = (cs_xdef_dof_context_t *)def->context;

        /* Values of the function are defined at the primal faces */
        if (cs_flag_test(cx->loc, cs_flag_primal_face))
          cx->func(cm->n_fc, cm->f_ids, true, /* dense output */
                   cx->input, fluxes);
        else
          bft_error(__FILE__, __LINE__, 0,
                    "%s: Invalid location for definition by DoFs.\n", __func__);

        cs_reco_cw_cell_vect_from_flux(cm, fluxes, vector_values);
        cs_nvec3(vector_values, eval);
      }
      break;

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
    case CS_XDEF_BY_VALUE:
      bft_error(__FILE__, __LINE__, 0,
                " %s: Type of definition not implemented yet.", __func__);
      break;

    default:
      bft_error(__FILE__, __LINE__, 0,
                " %s: Incompatible type of definition.", __func__);
      break;

    } /* Type of definition */

  }
  else {

    assert(adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR);
    assert(def->dim == 3);

    switch (def->type) {

    case CS_XDEF_BY_VALUE:
      {
        const cs_real_t  *constant_val = (cs_real_t *)def->context;

        cs_nvec3(constant_val, eval);
      }
      break; /* definition by value */

    case CS_XDEF_BY_ARRAY:        /* P0 approximation */
      cs_xdef_cw_eval_vector_at_xyz_by_array(cm, 1, xyz, time_eval,
                                             def->context,
                                             vector_values);
      cs_nvec3(vector_values, eval);
      break;

    case CS_XDEF_BY_FIELD:
      if (adv->vtx_field_id < 0 && adv->cell_field_id < 0)
        bft_error(__FILE__, __LINE__, 0,
                  "%s: This support is not available for this function.\n",
                  __func__);

      cs_xdef_cw_eval_vector_at_xyz_by_field(cm, 1, xyz, time_eval,
                                             def->context,
                                             vector_values);
      cs_nvec3(vector_values, eval);
      break;

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
      cs_xdef_cw_eval_at_xyz_by_analytic(cm, 1, xyz, time_eval, def->context,
                                         vector_values);
      cs_nvec3(vector_values, eval);
      break;

    default:
      bft_error(__FILE__, __LINE__, 0, " %s: Incompatible type of definition.",
                __func__);
      break;

    } /* Type of definition */

  } /* Velocity vector */
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the mean-value of the vector-valued field related to the
 *         advection field inside each cell
 *
 * \param[in]      adv           pointer to a cs_adv_field_t structure
 * \param[in]      time_eval     physical time at which one evaluates the term
 * \param[in, out] cell_values   array of values at cell centers
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_in_cells(const cs_adv_field_t   *adv,
                            cs_real_t               time_eval,
                            cs_real_t              *cell_values)
{
  if (adv == NULL)
    return;

  const cs_cdo_quantities_t  *cdoq = cs_cdo_quant;

  cs_xdef_t  *def = adv->definition;
  assert(def != NULL);  /* Sanity checks */

  if (adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX) {

    switch (def->type) {

    case CS_XDEF_BY_ARRAY:
      {
        cs_xdef_array_context_t  *ai = (cs_xdef_array_context_t *)def->context;

        if (cs_flag_test(ai->loc, cs_flag_dual_face_byc)) {

          assert(ai->index == cs_cdo_connect->c2e->idx);

#         pragma omp parallel for if (cdoq->n_cells > CS_THR_MIN)
          for (cs_lnum_t c_id = 0; c_id < cdoq->n_cells; c_id++)
            cs_reco_dfbyc_at_cell_center(c_id,
                                         cs_cdo_connect->c2e,
                                         cdoq,
                                         ai->values,
                                         cell_values + 3*c_id);

        }
        else if (cs_flag_test(ai->loc, cs_flag_primal_face)) {

          assert(adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX);

          cs_reco_cell_vectors_by_face_dofs(cs_cdo_connect->c2f,
                                            cdoq,
                                            ai->values,
                                            cell_values);

        }
        else
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid location for array", __func__);

      }
      break;

    case CS_XDEF_BY_FIELD:
      {
        const cs_field_t  *field = (cs_field_t *)def->context;
        assert(field != NULL);
        assert(field->dim == 1);
        const cs_mesh_location_type_t  loc_type =
          cs_mesh_location_get_type(field->location_id);

        if (loc_type == CS_MESH_LOCATION_INTERIOR_FACES) {

          /* Sanity checks */
          assert(adv->bdy_field_id > -1);
          cs_field_t  *b_mflx_fld = cs_field_by_id(adv->bdy_field_id);
          assert(b_mflx_fld != NULL);

          cs_reco_cell_vectors_by_ib_face_dofs(cs_cdo_connect->c2f,
                                               cdoq,
                                               field->val,
                                               b_mflx_fld->val,
                                               cell_values);
        }
        else
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid support for the input field", __func__);
      }
      break;

    case CS_XDEF_BY_DOF_FUNCTION:   /* P0 approximation in each cell */
      {
        cs_real_t  *fluxes = cs_equation_get_tmpbuf();
        cs_xdef_dof_context_t  *cx = (cs_xdef_dof_context_t *)def->context;

        if (cs_flag_test(cx->loc, cs_flag_primal_face) == false)
          bft_error(__FILE__, __LINE__, 0,
                    "%s: Invalid location for definition by DoFs.\n", __func__);
        assert(cs_equation_get_tmpbuf_size() <= (size_t)cdoq->n_faces);

        /* Values of the function are defined at the primal faces */
        cx->func(cdoq->n_faces, NULL, true, /* dense output */
                 cx->input, fluxes);

        cs_reco_cell_vectors_by_face_dofs(cs_cdo_connect->c2f,
                                          cdoq,
                                          fluxes,
                                          cell_values);
      }
      break;

    default:
      bft_error(__FILE__, __LINE__, 0, "%s: Incompatible type of definition.",
                __func__);
      break;

    } /* Type of definition */

  }
  else {

    assert(adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR);

    switch (def->type) {

    case CS_XDEF_BY_VALUE:
      {
        const cs_real_t  *constant_val = (cs_real_t *)def->context;

#       pragma omp parallel for if (cdoq->n_cells > CS_THR_MIN)
        for (cs_lnum_t i = 0; i < cdoq->n_cells; i++) {
          cell_values[3*i  ] = constant_val[0];
          cell_values[3*i+1] = constant_val[1];
          cell_values[3*i+2] = constant_val[2];
        }
      }
      break; /* definition by value */

    case CS_XDEF_BY_ARRAY:
      {
        cs_xdef_array_context_t  *ai = (cs_xdef_array_context_t *)def->context;

        assert(ai->stride == 3);
        if (!cs_flag_test(ai->loc, cs_flag_primal_cell))
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid location for array", __func__);

        memcpy(cell_values, ai->values,
               ai->stride*cdoq->n_cells*sizeof(cs_real_t));
      }
      break; /* definition by array */

    case CS_XDEF_BY_FIELD:
      {
        const cs_field_t  *field = (cs_field_t *)def->context;
        assert(field != NULL);
        const cs_mesh_location_type_t  loc_type =
          cs_mesh_location_get_type(field->location_id);

        switch(loc_type) {

        case CS_MESH_LOCATION_CELLS:
          assert(field->dim == 3);
          /* If not the field associated to the advection field */
          if (field->id != adv->cell_field_id)
            memcpy(cell_values, field->val, 3*cdoq->n_cells*sizeof(cs_real_t));
          break;

        case CS_MESH_LOCATION_VERTICES:
          assert(field->dim == 3);
          cs_reco_vect_pv_at_cell_centers(cs_cdo_connect->c2v,
                                          cdoq,
                                          field->val,
                                          cell_values);
          break;

        default:
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid support for the input field", __func__);
          break;

        } /* Switch on location type */

      }
      break; /* definition by field */

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
      cs_evaluate_average_on_cells_by_analytic(def, time_eval, cell_values);
      break; /* definition by analytic */

    default:
      bft_error(__FILE__, __LINE__, 0, " %s: Incompatible type of definition.",
                __func__);
      break;

    } /* Type of definition */

  } /* Velocity vector */
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the value of the advection field at vertices
 *
 * \param[in]      adv          pointer to a cs_adv_field_t structure
 * \param[in]      time_eval    physical time at which one evaluates the term
 * \param[in, out] vtx_values   array storing the results
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_at_vertices(const cs_adv_field_t    *adv,
                               cs_real_t                time_eval,
                               cs_real_t               *vtx_values)
{
  if (adv == NULL)
    return;

  const cs_cdo_quantities_t  *cdoq = cs_cdo_quant;
  const cs_cdo_connect_t  *connect = cs_cdo_connect;

  cs_xdef_t  *def = adv->definition;

  if (adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX) {

    /* Compute the vector-valued vector at each vertex */
    _compute_adv_vector_at_vertices(cdoq, connect, def, vtx_values);

    /* Synchronization of values at vertices */
    if (connect->interfaces[CS_CDO_CONNECT_VTX_SCAL] != NULL)
      cs_interface_set_sum(connect->interfaces[CS_CDO_CONNECT_VTX_SCAL],
                           cdoq->n_vertices,
                           3,             /* stride */
                           true,          /* = interlace */
                           CS_REAL_TYPE,
                           vtx_values);

    cs_real_t  *dual_vol = NULL;
    BFT_MALLOC(dual_vol, cdoq->n_vertices, cs_real_t);
    cs_cdo_quantities_compute_dual_volumes(cdoq, connect->c2v, dual_vol);

    if (connect->interfaces[CS_CDO_CONNECT_VTX_SCAL] != NULL)
      cs_interface_set_sum(connect->interfaces[CS_CDO_CONNECT_VTX_SCAL],
                           cdoq->n_vertices,
                           1,             /* stride */
                           true,          /* = interlace */
                           CS_REAL_TYPE,
                           dual_vol);

#   pragma omp parallel for if (cdoq->n_vertices > CS_THR_MIN)
    for (cs_lnum_t v_id = 0; v_id < cdoq->n_vertices; v_id++) {
      const cs_real_t  invvol = 1./dual_vol[v_id];
      for (int k = 0; k < 3; k++)
        vtx_values[3*v_id+k] *= invvol;
    }

    BFT_FREE(dual_vol);

  }
  else {

    assert(adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR);

    switch (def->type) {

    case CS_XDEF_BY_VALUE:
      {
        const cs_real_t  *constant_val = (cs_real_t *)def->context;

#     pragma omp parallel for if (cdoq->n_vertices > CS_THR_MIN)
        for (cs_lnum_t i = 0; i < cdoq->n_vertices; i++) {
          vtx_values[3*i  ] = constant_val[0];
          vtx_values[3*i+1] = constant_val[1];
          vtx_values[3*i+2] = constant_val[2];
        }

      }
      break; /* definition by value */

    case CS_XDEF_BY_ARRAY:
      {
        cs_xdef_array_context_t  *ai =
          (cs_xdef_array_context_t *)def->context;
        assert(ai->stride == 3);

        if (cs_flag_test(ai->loc, cs_flag_primal_vtx))
          memcpy(vtx_values, ai->values, 3*cdoq->n_vertices*sizeof(cs_real_t));

        else if (cs_flag_test(ai->loc, cs_flag_primal_cell))
          cs_reco_vect_pv_from_pc(connect->c2v, cdoq, ai->values, vtx_values);

        else
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid location for array", __func__);
      }
      break; /* definition by array */

    case CS_XDEF_BY_FIELD:
      {
        cs_field_t  *field = (cs_field_t *)def->context;
        assert(field != NULL);
        assert(field->dim == 3);

        switch(cs_mesh_location_get_type(field->location_id)) {

        case CS_MESH_LOCATION_CELLS:
          cs_reco_vect_pv_from_pc(cs_cdo_connect->c2v,
                                  cdoq,
                                  field->val,
                                  vtx_values);
          break;

        case CS_MESH_LOCATION_VERTICES:
          /* If not the field associated to the advection field */
          if (field->id != adv->vtx_field_id)
            memcpy(vtx_values, field->val,
                   3*cdoq->n_vertices*sizeof(cs_real_t));
          break;

        default:
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid case for the input field", __func__);
          break;

        }

      }
      break; /* definition by field */

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
      cs_evaluate_potential_at_vertices_by_analytic(def,
                                                    time_eval,
                                                    cdoq->n_vertices,
                                                    NULL,
                                                    vtx_values);
      break; /* definition by analytic */

    default:
      bft_error(__FILE__, __LINE__, 0, " %s: Incompatible type of definition.",
                __func__);
      break;

    } /* Type of definition */

  } /* velocity vector */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the value of the normal flux of the advection field
 *         across the boundary faces
 *
 * \param[in]      adv          pointer to a cs_adv_field_t structure
 * \param[in]      time_eval    physical time at which one evaluates the term
 * \param[in, out] flx_values   array storing the results
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_across_boundary(const cs_adv_field_t  *adv,
                                   cs_real_t              time_eval,
                                   cs_real_t             *flx_values)
{
  if (adv == NULL)
    return;
  if (adv->status & CS_ADVECTION_FIELD_LEGACY_FV)
    return;

  const cs_cdo_quantities_t  *cdoq = cs_cdo_quant;
  const cs_lnum_t  n_b_faces = cdoq->n_b_faces;
  const cs_lnum_t  n_i_faces = cdoq->n_i_faces;
  const cs_xdef_t  *def = adv->definition;

  if (adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR) {

    assert(def->dim == 3);
    if (adv->n_bdy_flux_defs > 0)
      bft_error(__FILE__, __LINE__, 0,
                "%s: Case where one expects a volume definition only.",
                __func__);

    /* No specific definition of the boundary flux should be set.
     * So, one uses the definition related to the volume */

    switch (def->type) {

    case CS_XDEF_BY_VALUE:
      {
        const cs_real_t  *constant_val = (cs_real_t *)def->context;

#       pragma omp parallel for if  (n_b_faces > CS_THR_MIN)
        for (cs_lnum_t i = 0; i < n_b_faces; i++)
          flx_values[i] = _dp3(cdoq->b_face_normal + 3*i, constant_val);

      }
      break;

    case CS_XDEF_BY_ARRAY:
      {
        cs_xdef_array_context_t *ai = (cs_xdef_array_context_t *)def->context;
        assert(ai->stride == 3);

        if (cs_flag_test(ai->loc, cs_flag_primal_face)) {

          /* Boundary faces come after interior faces */
          const cs_real_t  *const bface_vel = ai->values + 3*n_i_faces;

#         pragma omp parallel for if  (n_b_faces > CS_THR_MIN)
          for (cs_lnum_t i = 0; i < n_b_faces; i++)
            flx_values[i] = _dp3(cdoq->b_face_normal + 3*i, bface_vel + 3*i);

        }
        else if (cs_flag_test(ai->loc, cs_flag_primal_cell)) {

          const cs_adjacency_t  *f2c = cs_cdo_connect->f2c;
          const cs_lnum_t  *const bf2c_ids = f2c->ids + 2*n_i_faces;
          const cs_real_t  *const cell_vel = ai->values;

#         pragma omp parallel for if  (n_b_faces > CS_THR_MIN)
          for (cs_lnum_t i = 0; i < n_b_faces; i++) {
            const cs_lnum_t  c_id = bf2c_ids[i];
            flx_values[i] = _dp3(cdoq->b_face_normal + 3*i, cell_vel + 3*c_id);
          }

        }
        else
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Incompatible location when defined by array.",
                    __func__);

      }
      break;

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
      {
        const cs_adjacency_t  *f2e = cs_cdo_connect->f2e;
        const cs_adjacency_t  *e2v = cs_cdo_connect->e2v;
        const cs_real_t  *xv = cdoq->vtx_coord;

        cs_quadrature_tria_integral_t
          *compute_integral = cs_quadrature_get_tria_integral(def->dim,
                                                              def->qtype);
        cs_xdef_analytic_context_t *ac =
          (cs_xdef_analytic_context_t *)def->context;

        for (cs_lnum_t i = 0; i < n_b_faces; i++) {

          const cs_lnum_t  f_id = n_i_faces + i;
          const cs_quant_t  pfq = cs_quant_set_face(f_id, cdoq);
          const cs_lnum_t  start_idx = f2e->idx[f_id];
          const cs_lnum_t  end_idx = f2e->idx[f_id+1];

          cs_real_t  val[3] = {0, 0, 0};

          switch (end_idx - start_idx) {

          case CS_TRIANGLE_CASE: /* Triangle: one-shot computation */
            {
              cs_lnum_t  v1, v2, v3;

              cs_connect_get_next_3_vertices(f2e->ids, e2v->ids, start_idx,
                                             &v1, &v2, &v3);

              compute_integral(time_eval,
                               xv + 3*v1, xv + 3*v2, xv + 3*v3, pfq.meas,
                               ac->func, ac->input, val);
            }
            break;

          default:
            for (cs_lnum_t j = start_idx; j < end_idx; j++) {

              const cs_lnum_t  _2e = 2*f2e->ids[j];
              const cs_real_t  *xv1 = xv + 3*e2v->ids[_2e];
              const cs_real_t  *xv2 = xv + 3*e2v->ids[_2e+1];
              const cs_real_t  tef_meas = cs_math_surftri(xv1, xv2, pfq.center);

              /* val is updated (+=) */
              compute_integral(time_eval, xv1, xv2, pfq.center, tef_meas,
                               ac->func, ac->input, val);

            } /* Loop on face edges */

          } /* End of switch */

          flx_values[i] = _dp3(pfq.unitv, val);

        } /* Loop on border faces */

      }
      break; /* definition by analytic */

    default:
      bft_error(__FILE__, __LINE__, 0, " %s: Incompatible type of definition.",
                __func__);
      break;

    } /* Type of definition */

  }   /* Velocity vector */
  else {

    assert(adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX);

    if (adv->n_bdy_flux_defs == 0) {

      switch (def->type) {

      case CS_XDEF_BY_ARRAY:
        {
          const cs_xdef_array_context_t *ac =
            (cs_xdef_array_context_t *)def->context;

          assert(ac->stride == 1);
          assert(def->meta & CS_FLAG_FULL_LOC); /* over the volume hence the
                                                   shift with n_i_faces */

          if (cs_flag_test(ac->loc, cs_flag_primal_face))
            memcpy(flx_values, ac->values + n_i_faces,
                   sizeof(cs_real_t)*n_b_faces);
        }
        break;

      case CS_XDEF_BY_DOF_FUNCTION:
        {
          cs_xdef_dof_context_t  *cx = (cs_xdef_dof_context_t *)def->context;

          if (cs_flag_test(cx->loc, cs_flag_primal_face) == false)
            bft_error(__FILE__, __LINE__, 0,
                      "%s: Invalid location for definition by DoFs.\n",
                      __func__);

          cs_lnum_t  *bface_ids = NULL;
          BFT_MALLOC(bface_ids, n_b_faces, cs_lnum_t);
          for (cs_lnum_t i = 0; i < n_b_faces; i++)
            bface_ids[i] = n_i_faces + i;

          cx->func(n_b_faces, bface_ids, true, /* dense output */
                   cx->input,
                   flx_values);

          BFT_FREE(bface_ids);
        }
        break;

      default:
        bft_error(__FILE__, __LINE__, 0,
                  "%s: Invalid type of definition in case of flux defined\n"
                  "on the whole domain. It should a definition by array.",
                  __func__);
      }

    }
    else {

      /* Consider the definition of the boundary flux for updating the field
         values */
      for (int def_id = 0; def_id < adv->n_bdy_flux_defs; def_id++) {

        const cs_xdef_t  *bdef = adv->bdy_flux_defs[def_id];
        const cs_zone_t  *z = cs_boundary_zone_by_id(bdef->z_id);

        assert(bdef->support == CS_XDEF_SUPPORT_BOUNDARY);
        assert(bdef->dim == 1);

        switch (bdef->type) {

        case CS_XDEF_BY_VALUE:
          {
            const cs_real_t  *constant_val = (cs_real_t *)bdef->context;

            if (z->elt_ids == NULL) {
              assert(z->n_elts == n_b_faces);
#           pragma omp parallel for if (n_b_faces > CS_THR_MIN)
              for (cs_lnum_t i = 0; i < n_b_faces; i++)
                flx_values[i] = constant_val[0];
            }
            else {
#           pragma omp parallel for if (z->n_elts > CS_THR_MIN)
              for (cs_lnum_t i = 0; i < z->n_elts; i++)
                flx_values[z->elt_ids[i]] = constant_val[0];
            }

          }
          break;

        case CS_XDEF_BY_ARRAY:
          {
            const cs_xdef_array_context_t *ac =
              (cs_xdef_array_context_t *)bdef->context;
            const cs_real_t  *val = ac->values;

            assert(ac->stride == 1);
            assert(bdef->meta & CS_FLAG_FULL_LOC || z->elt_ids == NULL);

            if (cs_flag_test(ac->loc, cs_flag_primal_face))
              memcpy(flx_values, val, sizeof(cs_real_t)*n_b_faces);

            else if (cs_flag_test(ac->loc, cs_flag_dual_closure_byf)) {

              for (cs_lnum_t bf_id = 0; bf_id < n_b_faces; bf_id++) {

                flx_values[bf_id] = 0;
                for (cs_lnum_t i = ac->index[bf_id]; i < ac->index[bf_id+1];
                     i++)
                  flx_values[bf_id] += val[i];

              } /* Loop on border faces */

            }
            else
              bft_error(__FILE__, __LINE__, 0, "%s: Invalid case.", __func__);

          }
          break; /* definition by array */

        case CS_XDEF_BY_FIELD:
          {
            cs_field_t  *field = (cs_field_t *)bdef->context;

            assert(field->dim == 1);
            assert(bdef->meta & CS_FLAG_FULL_LOC || z->elt_ids == NULL);

            if (field->location_id ==
                cs_mesh_location_get_id_by_name(N_("boundary faces")))
              memcpy(flx_values, field->val, sizeof(cs_real_t)*n_b_faces);
            else
              bft_error(__FILE__, __LINE__, 0, " %s: Invalid case.", __func__);

          }
          break; /* definition by field */

        case CS_XDEF_BY_ANALYTIC_FUNCTION:
          {
            cs_xdef_analytic_context_t  *ac =
              (cs_xdef_analytic_context_t *)bdef->context;

            ac->func(time_eval,
                     z->n_elts, z->elt_ids, cdoq->b_face_center,
                     false,   /* compacted output ? */
                     ac->input,
                     flx_values);
          }
          break; /* definition by analytic */

        default:
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Incompatible type of definition.", __func__);
          break;

        } /* Type of definition */

      } /*  Loop on boundary definitions */

    } /* There is at least one boundary definition */

  } /* Scalar-valued flux */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the value of the normal flux of the advection field across
 *         a boundary face f (cellwise version)
 *
 * \param[in] time_eval   physical time at which one evaluates the term
 * \param[in] f           face id in the cellwise numbering
 * \param[in] cm          pointer to a cs_cell_mesh_t structure
 * \param[in] adv         pointer to a cs_adv_field_t structure
 *
 * \return  the normal boundary flux for the face f
 */
/*----------------------------------------------------------------------------*/

cs_real_t
cs_advection_field_cw_boundary_face_flux(const cs_real_t          time_eval,
                                         const short int          f,
                                         const cs_cell_mesh_t    *cm,
                                         const cs_adv_field_t    *adv)
{
  cs_real_t  f_flux = 0.;

  if (adv == NULL)
    return f_flux;

  const cs_quant_t  pfq = cm->face[f];
  const cs_lnum_t  bf_id = cm->f_ids[f] - cm->bface_shift;

  assert(bf_id > -1);
  assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_PFQ));

  if (adv->bdy_field_id > -1) { /* Use the current computed advection field
                                   across the boundary */

    cs_field_t  *fld = cs_field_by_id(adv->bdy_field_id);

    return fld->val[bf_id];

  }

  const cs_xdef_t  *def = adv->definition;

  if (adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR) {

    /* No specific definition of the boundary flux. Use the definition related
       to the volume */
    assert(adv->n_bdy_flux_defs == 0);
    assert(def->dim == 3);

    switch (def->type) {

    case CS_XDEF_BY_VALUE:
      {
        const cs_real_t  *constant_val = (cs_real_t *)def->context;

        f_flux = pfq.meas * _dp3(pfq.unitv, constant_val);
      }
      break;

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
      {
        cs_real_t  adv_val[3] = {0, 0, 0};
        cs_quadrature_tria_integral_t *compute_integral =
          cs_quadrature_get_tria_integral(def->dim, def->qtype);
        cs_xdef_analytic_context_t *ac =
          (cs_xdef_analytic_context_t *)def->context;

        assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_PV  | CS_FLAG_COMP_FEQ |
                             CS_FLAG_COMP_EV));

        const cs_lnum_t  start_idx = cm->f2e_idx[f];
        const cs_lnum_t  end_idx = cm->f2e_idx[f+1];
        const short int  n_vf = end_idx - start_idx;  /* #vertices (=#edges) */
        const short int  *f2e_ids = cm->f2e_ids + start_idx;

        switch (n_vf) {

        case CS_TRIANGLE_CASE: /* Triangle: one-shot computation */
          {
            short int  v1, v2, v3;
            cs_cell_mesh_get_next_3_vertices(f2e_ids, cm->e2v_ids,
                                             &v1, &v2, &v3);

            compute_integral(time_eval,
                             cm->xv + 3*v1, cm->xv + 3*v2, cm->xv + 3*v3,
                             pfq.meas, ac->func, ac->input,
                             adv_val);
          }
          break;

        default:
          {
            const double  *tef = cm->tef + start_idx;

            for (short int e = 0; e < n_vf; e++) { /* Loop on face edges */

              const short int  _2e = 2*f2e_ids[e];
              const cs_real_t  *xv1 = cm->xv + 3*cm->e2v_ids[_2e];
              const cs_real_t  *xv2 = cm->xv + 3*cm->e2v_ids[_2e+1];

              /* adv_val is updated (+=) */
              compute_integral(time_eval, xv1, xv2, pfq.center, tef[e],
                               ac->func, ac->input,
                               adv_val);
            } /* Loop on face edges */

          }

        } /* End of switch (n_vf) */

        f_flux = _dp3(pfq.unitv, adv_val);

      }
      break; /* definition by analytic */

    default:
      bft_error(__FILE__, __LINE__, 0, " %s: Incompatible type of definition.",
                __func__);
      break;

    } /* Type of definition */

  }
  else {

    assert(adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX);
    if (adv->n_bdy_flux_defs == 0) {

      switch (def->type) {

      case CS_XDEF_BY_ARRAY:
        {
          const cs_xdef_array_context_t  *ac =
            (cs_xdef_array_context_t *)def->context;

          assert(ac->stride == 1);
          if (cs_flag_test(ac->loc, cs_flag_primal_face))
            f_flux = ac->values[cm->f_ids[f]];
          else
            bft_error(__FILE__, __LINE__, 0,
                      "%s: Invalid array support.", __func__);
        }
        break;

      case CS_XDEF_BY_DOF_FUNCTION:
        {
          cs_xdef_dof_context_t  *cx = (cs_xdef_dof_context_t *)def->context;

          if (cs_flag_test(cx->loc, cs_flag_primal_face) == false)
            bft_error(__FILE__, __LINE__, 0,
                      "%s: Invalid location for definition by DoFs.\n",
                      __func__);

          /* Values of the function are defined at the primal faces */
          cx->func(1, cm->f_ids + f, true, /* dense output */
                   cx->input,
                   &f_flux);
        }
        break;

      default:
        bft_error(__FILE__, __LINE__, 0,
                  "%s: Invalid type of definition.", __func__);

      } /* def->type */

    }
    else { /* n_bdy_flux_defs > 0 */

      const cs_xdef_t  *bdef = (adv->bdy_def_ids == NULL) ?
        adv->bdy_flux_defs[0] : adv->bdy_flux_defs[adv->bdy_def_ids[bf_id]];

      assert(bdef != NULL);
      assert(bdef->support == CS_XDEF_SUPPORT_BOUNDARY);

      switch (bdef->type) {

      case CS_XDEF_BY_VALUE:
        {
          const cs_real_t  *constant_val = (cs_real_t *)bdef->context;
          f_flux = constant_val[0];
        }
        break;

      case CS_XDEF_BY_ARRAY:
        {
          const cs_xdef_array_context_t  *ac =
            (cs_xdef_array_context_t *)bdef->context;

          assert(ac->stride == 1);
          if (cs_flag_test(ac->loc, cs_flag_primal_face))
            f_flux = ac->values[bf_id];

          else if (cs_flag_test(ac->loc, cs_flag_dual_closure_byf)) {

            const cs_adjacency_t  *bf2v = cs_cdo_connect->bf2v;
            for (cs_lnum_t i = bf2v->idx[bf_id]; i < bf2v->idx[bf_id+1]; i++)
              f_flux += ac->values[i];

          }
          else
            bft_error(__FILE__, __LINE__, 0, " %s: Invalid support", __func__);

        }
        break; /* definition by array */

      case CS_XDEF_BY_FIELD:
        {
          cs_field_t  *field = (cs_field_t *)bdef->context;
          assert(field->dim == 1);

          if (field->location_id ==
              cs_mesh_location_get_id_by_name(N_("boundary faces")))
            f_flux = field->val[bf_id];
          else
            bft_error(__FILE__, __LINE__, 0, " %s: Invalid case.", __func__);

        }
        break; /* definition by field */

      case CS_XDEF_BY_ANALYTIC_FUNCTION:
        {
          cs_xdef_analytic_context_t *ac =
            (cs_xdef_analytic_context_t *)def->context;

          ac->func(time_eval,
                   1, NULL, pfq.center,
                   true,   /* compacted output ? */
                   ac->input,
                   &f_flux);
        }
        break; /* definition by analytic */

      default:
        bft_error(__FILE__, __LINE__, 0,
                  " %s: Incompatible type of definition.", __func__);
        break;

      } /* Type of definition */

    } /* There is at least one definition of the normal boundary flux */

  } /* Scalar-valued flux */

  return f_flux;
}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the value of the normal flux of the advection field across
 *         the closure of the dual cell related to each vertex belonging to the
 *         boundary face f
 *
 * \param[in]      cm         pointer to a cs_cell_mesh_t structure
 * \param[in]      adv        pointer to a cs_adv_field_t structure
 * \param[in]      f          face id in the cellwise numbering
 * \param[in]      time_eval  physical time at which one evaluates the term
 * \param[in, out] fluxes     normal boundary flux for each vertex of the face
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_cw_boundary_f2v_flux(const cs_cell_mesh_t   *cm,
                                        const cs_adv_field_t   *adv,
                                        short int               f,
                                        cs_real_t               time_eval,
                                        cs_real_t              *fluxes)
{
  if (adv == NULL)
    return;

  if (fluxes == NULL)
    bft_error(__FILE__, __LINE__, 0,
              " %s: Array of fluxes should be allocated before the call.",
              __func__);

  /* Reset flux values */
  for (short int v = 0; v < cm->n_vc; v++) fluxes[v] = 0;

  const cs_quant_t  pfq = cm->face[f];
  const cs_lnum_t  bf_id = cm->f_ids[f] - cm->bface_shift;

  assert(bf_id > -1);

  /* If the boundary flux has been computed, used this array expected in case of
     analytical definition on the boundary */
  if (adv->bdy_field_id > -1 && adv->n_bdy_flux_defs == 0) {

    cs_field_t  *fld = cs_field_by_id(adv->bdy_field_id);

    _cw_fill_uniform_boundary_flux(cm, f, fld->val[bf_id], fluxes);

  }
  else if (adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX) {

    if (adv->n_bdy_flux_defs == 0) {

      const cs_xdef_t  *def = adv->definition;

      if (def->type != CS_XDEF_BY_ARRAY)
        bft_error(__FILE__, __LINE__, 0,
                  "%s: Invalid type of definition", __func__);

      const cs_xdef_array_context_t  *ac =
        (cs_xdef_array_context_t *)def->context;

      assert(cs_flag_test(ac->loc, cs_flag_primal_face));
      _cw_fill_uniform_boundary_flux(cm, f, ac->values[bf_id], fluxes);

    }
    else {

      assert(cs_eflag_test(cm->flag,
                           CS_FLAG_COMP_PFQ | CS_FLAG_COMP_FEQ |
                           CS_FLAG_COMP_EV  | CS_FLAG_COMP_FE));

      /* Consider the definition of the boundary flux to set the values */
      const cs_xdef_t  *def = (adv->bdy_def_ids == NULL) ?
        adv->bdy_flux_defs[0] : adv->bdy_flux_defs[adv->bdy_def_ids[bf_id]];

      switch (def->type) {

      case CS_XDEF_BY_VALUE:
        {
          const cs_real_t  *constant_val = (cs_real_t *)def->context;

          _cw_fill_uniform_boundary_flux(cm, f, constant_val[0], fluxes);
        }
        break; /* by value */

      case CS_XDEF_BY_ARRAY:
        {
          const cs_xdef_array_context_t  *ac
            = (cs_xdef_array_context_t *)def->context;

          assert(ac->stride == 1);
          if (cs_flag_test(ac->loc, cs_flag_primal_face))
            _cw_fill_uniform_boundary_flux(cm, f, ac->values[bf_id], fluxes);

          else if (cs_flag_test(ac->loc, cs_flag_dual_closure_byf)) {

            const cs_adjacency_t  *bf2v = cs_cdo_connect->bf2v;
            for (cs_lnum_t i = bf2v->idx[bf_id]; i < bf2v->idx[bf_id+1]; i++) {
              const short int v = cs_cell_mesh_get_v(bf2v->ids[i], cm);
              assert(v > -1 && v != cm->n_vc);
              fluxes[v] += ac->values[i];
            }

          }
          else
            bft_error(__FILE__, __LINE__, 0, " %s: Invalid support", __func__);
        }
        break; /* by_array */

      case CS_XDEF_BY_FIELD:
        {
          cs_field_t  *field = (cs_field_t *)def->context;
          assert(field->dim == 1);

          if (cs_mesh_location_get_type(field->location_id) ==
              CS_MESH_LOCATION_BOUNDARY_FACES)
            _cw_fill_uniform_boundary_flux(cm, f, field->val[bf_id], fluxes);
          else
            bft_error(__FILE__, __LINE__, 0, " %s: Invalid support", __func__);
        }
        break; /* definition by field */

      case CS_XDEF_BY_ANALYTIC_FUNCTION:
        {
          cs_real_t  f_flux = 0;
          cs_xdef_analytic_context_t *ac =
            (cs_xdef_analytic_context_t *)def->context;

          ac->func(time_eval,
                   1, NULL, pfq.center,
                   true,   /* compacted output ? */
                   ac->input,
                   &f_flux);

          _cw_fill_uniform_boundary_flux(cm, f, f_flux, fluxes);
        }
        break; /* definition by analytic */

      default:
        bft_error(__FILE__, __LINE__, 0,
                  " %s: Invalid type of definition", __func__);
        break;

      } /* End of switch on the type of definition */

    }

  }
  else {

    assert(adv->n_bdy_flux_defs == 0);
    assert(adv->bdy_def_ids == NULL);
    assert(adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR);

    /* Implicit definition of the boundary flux from the definition of the
       advection field inside the computational domain */
    cs_xdef_t  *def = adv->definition;

    assert(cs_eflag_test(cm->flag,
                         CS_FLAG_COMP_PFQ | CS_FLAG_COMP_FEQ |
                         CS_FLAG_COMP_EV  | CS_FLAG_COMP_FE));

    switch (def->type) {

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
      {
        cs_quadrature_tria_integral_t
          *compute_integral = cs_quadrature_get_tria_integral(def->dim,
                                                              def->qtype);
        cs_xdef_analytic_context_t *ac =
          (cs_xdef_analytic_context_t *)def->context;

        const short int end_idx = cm->f2e_idx[f+1];
        const short int start_idx = cm->f2e_idx[f];
        const short int n_ef = end_idx - start_idx; /* #vertices (= #edges) */
        const short int *f2e_ids = cm->f2e_ids + start_idx;
        const double  *tef = cm->tef + start_idx;

        for (short int e = 0; e < n_ef; e++) { /* Loop on face edges */

          const short int  _2e = 2*f2e_ids[e];
          const short int  v1 = cm->e2v_ids[_2e];
          const short int  v2 = cm->e2v_ids[_2e+1];

          cs_real_t  val[3] = {0, 0, 0};

          /* val is updated (+=) */
          compute_integral(time_eval,
                           cm->xv + 3*v1, cm->xv + 3*v2, pfq.center, tef[e],
                           ac->func, ac->input, val);

          const double  e_contrib = 0.5*_dp3(pfq.unitv, val);

          fluxes[v1] += e_contrib;
          fluxes[v2] += e_contrib;

        } /* Loop on face edges */
      }
      break;

    case CS_XDEF_BY_VALUE:
      {
        const cs_real_t  *constant_val = (cs_real_t *)def->context;
        const cs_real_t  half_face_density = 0.5*_dp3(pfq.unitv, constant_val);

        for (short int i = cm->f2e_idx[f]; i < cm->f2e_idx[f+1]; i++) {

          const short int  eshift = 2*cm->f2e_ids[i];
          const short int  v0 = cm->e2v_ids[eshift];
          const short int  v1 = cm->e2v_ids[eshift+1];
          const double  weighted_flux = half_face_density * cm->tef[i];

          fluxes[v0] += weighted_flux;
          fluxes[v1] += weighted_flux;

        } /* Loop on face edges */

      }
      break;

    default:
      bft_error(__FILE__, __LINE__, 0,
                "%s: Invalid type of definition", __func__);
      break;

    } /* End of switch */

  } /* n_bdy_flux_defs == 0 && CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the value of the flux of the advection field across the
 *         the (primal) faces of a cell
 *
 * \param[in]      cm         pointer to a cs_cell_mesh_t structure
 * \param[in]      adv        pointer to a cs_adv_field_t structure
 * \param[in]      time_eval  physical time at which one evaluates the term
 * \param[in, out] fluxes     array of values attached to primal faces of a cell
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_cw_face_flux(const cs_cell_mesh_t       *cm,
                                const cs_adv_field_t       *adv,
                                cs_real_t                   time_eval,
                                cs_real_t                  *fluxes)
{
  if (adv == NULL)
    return;

  if (fluxes == NULL) {
    bft_error(__FILE__, __LINE__, 0,
              " %s: The array of local fluxes should be already allocated.",
              __func__);
    return;
  }

  cs_xdef_t  *def = adv->definition;
  assert(def != NULL); /* Sanity check */

  if (adv->status & CS_ADVECTION_FIELD_TYPE_SCALAR_FLUX) {

    switch (def->type) {

    case CS_XDEF_BY_ARRAY:
      {
        const cs_real_t  *xdef_fluxes = cs_xdef_get_array(def);
        assert(fluxes != NULL);
        for (short int f = 0; f < cm->n_fc; f++)
          fluxes[f] = xdef_fluxes[cm->f_ids[f]];

      }
      break;

    case CS_XDEF_BY_FIELD:
      {
        cs_field_t  *fld = (cs_field_t *)def->context;

        /* Sanity check */
        assert(cs_flag_test(cm->flag, CS_FLAG_COMP_PFQ));
        assert(cs_mesh_location_get_type(fld->location_id)
               == CS_MESH_LOCATION_INTERIOR_FACES);
        assert(adv->bdy_field_id > -1);

        cs_field_t  *b_mflx_fld = cs_field_by_id(adv->bdy_field_id);
        assert(b_mflx_fld != NULL);

        const cs_real_t  *b_flux = b_mflx_fld->val;
        const cs_real_t  *i_flux = fld->val;

        /* Loop on cell faces */
        for (short int f = 0; f < cm->n_fc; f++) {

          const cs_lnum_t  f_id = cm->f_ids[f];
          if (cm->f_ids[f] < cm->bface_shift) /* Interior face */
            fluxes[f] = i_flux[f_id];
          else
            fluxes[f] = b_flux[f_id - cm->bface_shift];

        } /* Loop on cell faces */
      }
      break;

    case CS_XDEF_BY_DOF_FUNCTION:
      {
        cs_xdef_dof_context_t  *cx = (cs_xdef_dof_context_t *)def->context;

        assert(cs_flag_test(cx->loc, cs_flag_primal_face));
        cx->func(cm->n_fc, cm->f_ids, true, /* dense_output */
                 cx->input,
                 fluxes);
      }
      break;

    default:
      bft_error(__FILE__, __LINE__, 0,
                "%s: Invalid type of definition", __func__);

    } /* End of switch def->type */

  }
  else {

    assert(adv->status & CS_ADVECTION_FIELD_TYPE_VELOCITY_VECTOR);
    assert(def->dim == 3);

    switch (def->type) {

    case CS_XDEF_BY_VALUE:
      {
        /* Loop on cell faces */
        for (short int f = 0; f < cm->n_fc; f++)
          cs_xdef_cw_eval_flux_by_val(cm, f, time_eval, def->context, fluxes);
      }
      break;

    case CS_XDEF_BY_ANALYTIC_FUNCTION:
      {
        assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_FEQ | CS_FLAG_COMP_PFQ));

        /* Loop on cell faces */
        for (short int f = 0; f < cm->n_fc; f++)
          cs_xdef_cw_eval_flux_by_analytic(cm, f, time_eval,
                                           def->context, def->qtype,
                                           fluxes);
      }
      break;

    case CS_XDEF_BY_ARRAY:
      {
        cs_nvec3_t  nv;
        cs_xdef_array_context_t  *ac = (cs_xdef_array_context_t *)def->context;

        assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_PFQ));
        assert(ac->values != NULL);

        /* Test if location has at least the pattern of the reference support */
        if (cs_flag_test(ac->loc, cs_flag_primal_face)) {

          for (short int f = 0; f < cm->n_fc; f++) {

            /* Retrieve the advection field:
               Switch to a cs_nvec3_t representation */
            cs_nvec3(ac->values + 3*cm->f_ids[f], &nv);

            fluxes[f] = nv.meas*cm->face[f].meas * _dp3(nv.unitv,
                                                        cm->face[f].unitv);

          } /* Loop on cell faces */

        }
        else if (cs_flag_test(ac->loc, cs_flag_primal_cell)) {

          /* Retrieve the advection field:
             Switch to a cs_nvec3_t representation */
          cs_nvec3(ac->values + 3*cm->c_id, &nv);

          /* Loop on cell faces */
          for (short int f = 0; f < cm->n_fc; f++)
            fluxes[f] = nv.meas*cm->face[f].meas * _dp3(nv.unitv,
                                                        cm->face[f].unitv);

        }
        else
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid support for evaluating the advection field %s"
                    " at the cell center of cell %ld.",
                    __func__, adv->name, (long)cm->c_id);
      }
      break; /* Definition by array */

    case CS_XDEF_BY_FIELD:
      {
        cs_field_t  *fld = (cs_field_t *)def->context;

        /* Sanity check */
        assert(cs_flag_test(cm->flag, CS_FLAG_COMP_PFQ));
        assert(cs_mesh_location_get_type(fld->location_id)
               == CS_MESH_LOCATION_CELLS);

        /* Retrieve the advection field:
           Switch to a cs_nvec3_t representation */
        cs_nvec3_t  nv;
        cs_nvec3(fld->val + 3*cm->c_id, &nv);

        /* Loop on cell faces */
        for (short int f = 0; f < cm->n_fc; f++)
          fluxes[f] = nv.meas*cm->face[f].meas * _dp3(nv.unitv,
                                                      cm->face[f].unitv);
      }
      break; /* Definition by field */

    default:
      bft_error(__FILE__, __LINE__, 0,
                "%s: Incompatible type of definition.", __func__);
      break;

    } /* def_type */

  } /* Velocity vector */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief  Compute the value of the flux of the advection field across the
 *         the dual faces of a cell
 *
 * \param[in]      cm         pointer to a cs_cell_mesh_t structure
 * \param[in]      adv        pointer to a cs_adv_field_t structure
 * \param[in]      time_eval  physical time at which one evaluates the term
 * \param[in, out] fluxes     array of values attached to dual faces of a cell
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_cw_dface_flux(const cs_cell_mesh_t     *cm,
                                 const cs_adv_field_t     *adv,
                                 cs_real_t                 time_eval,
                                 cs_real_t                *fluxes)
{
  if (adv == NULL)
    return;

  if (fluxes == NULL) {
    bft_error(__FILE__, __LINE__, 0,
              " fluxes array should be allocated before the call.");
    return;
  }

  cs_xdef_t  *def = adv->definition;

  /* Sanity checks */
  assert(def != NULL);

  switch (def->type) {

  case CS_XDEF_BY_VALUE:
    {
      /* Retrieve the advection field: Switch to a cs_nvec3_t representation */
      cs_real_3_t  cell_vector;
      cs_xdef_cw_eval_vector_by_val(cm, time_eval, def->context, cell_vector);
      cs_nvec3_t  nv;
      cs_nvec3(cell_vector, &nv);

      /* Sanity check */
      assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_DFQ | CS_FLAG_COMP_PEQ));

      /* Loop on cell edges */
      for (short int e = 0; e < cm->n_ec; e++)
        fluxes[e] = nv.meas * cm->dface[e].meas * _dp3(nv.unitv,
                                                       cm->dface[e].unitv);
    }
    break;

  case CS_XDEF_BY_ANALYTIC_FUNCTION:
    {
      assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_PEQ | CS_FLAG_COMP_SEF |
                           CS_FLAG_COMP_PFQ));

      cs_quadrature_type_t  qtype = cs_xdef_get_quadrature(def);

      /* Reset fluxes across dual faces */
      memset(fluxes, 0, cm->n_ec*sizeof(cs_real_t));

      switch (qtype) {

      case CS_QUADRATURE_NONE:
      case CS_QUADRATURE_BARY:
      case CS_QUADRATURE_BARY_SUBDIV:
        {
          cs_real_3_t  xg, adv_xg;

          /* Loop on cell faces */
          for (short int f = 0; f < cm->n_fc; f++) {

            const cs_quant_t  face = cm->face[f];
            const cs_real_3_t  xfc = { cm->xc[0] + face.center[0],
                                       cm->xc[1] + face.center[1],
                                       cm->xc[2] + face.center[2] };

            for (short int i = cm->f2e_idx[f]; i < cm->f2e_idx[f+1]; i++) {

              const short int  e = cm->f2e_ids[i];
              const cs_quant_t  edge = cm->edge[e];
              const cs_nvec3_t  sefc = cm->sefc[i];

              for (int k = 0; k < 3; k++)
                xg[k] = cs_math_1ov3 * (xfc[k] + edge.center[k]);

              cs_xdef_cw_eval_at_xyz_by_analytic(cm,
                                                 1, (const cs_real_t *)xg,
                                                 time_eval,
                                                 def->context,
                                                 (cs_real_t *)adv_xg);

              fluxes[e] += sefc.meas * _dp3(adv_xg, sefc.unitv);

            } /* Loop on face edges */

          } /* Loop on cell faces */
        }
        break; /* quadrature NONE, BARY or BARY_SUBDIV */

      case CS_QUADRATURE_HIGHER:
        {
          /* 4 Gauss points by triangle spanned by x_c, x_f, x_e */
          cs_real_t  w[4], eval[12];
          cs_real_3_t  gpts[4];

          /* Loop on cell faces */
          for (short int f = 0; f < cm->n_fc; f++) {

            const cs_quant_t  face = cm->face[f];

            for (short int i = cm->f2e_idx[f]; i < cm->f2e_idx[f+1]; i++) {

              const short int  e = cm->f2e_ids[i];
              const cs_quant_t  edge = cm->edge[e];
              const cs_nvec3_t  sefc = cm->sefc[i];

              cs_quadrature_tria_4pts(edge.center, face.center, cm->xc,
                                      sefc.meas,
                                      gpts, w);

              cs_xdef_cw_eval_at_xyz_by_analytic(cm,
                                                 4, (const cs_real_t *)gpts,
                                                 time_eval,
                                                 def->context,
                                                 eval);

              cs_real_t  add = 0;
              for (int p = 0; p < 4; p++)
                add += w[p] * _dp3(eval + 3*p, sefc.unitv);
              fluxes[e] += add;

            } /* Loop on face edges */

          } /* Loop on cell faces */
        }
        break; /* CS_QUADRATURE_HIGHER */

      case CS_QUADRATURE_HIGHEST:
        {
          /* 7 Gauss points by triangle spanned by x_c, x_f, x_e */
          cs_real_t  w[7], eval[21];
          cs_real_3_t  gpts[7];

          /* Loop on cell faces */
          for (short int f = 0; f < cm->n_fc; f++) {

            const cs_quant_t  face = cm->face[f];

            for (short int i = cm->f2e_idx[f]; i < cm->f2e_idx[f+1]; i++) {

              const short int  e = cm->f2e_ids[i];
              const cs_quant_t  edge = cm->edge[e];
              const cs_nvec3_t  sefc = cm->sefc[i];

              cs_quadrature_tria_7pts(edge.center, face.center, cm->xc,
                                      sefc.meas,
                                      gpts, w);

              cs_xdef_cw_eval_at_xyz_by_analytic(cm,
                                                 7, (const cs_real_t *)gpts,
                                                 time_eval,
                                                 def->context,
                                                 eval);

              cs_real_t  add = 0;
              for (int p = 0; p < 7; p++)
                add += w[p] * _dp3(eval + 3*p, sefc.unitv);
              fluxes[e] += add;

            } /* Loop on face edges */

          } /* Loop on cell faces */
        }
        break; /* CS_QUADRATURE_HIGHEST */

      default:
        bft_error(__FILE__, __LINE__, 0,
                  " %s: Invalid type of quadrature.", __func__);
        break;

      }  /* Switch type of quadrature */

    }  /* Definition by analytic function */
    break;

  case CS_XDEF_BY_ARRAY:
    {
      cs_xdef_array_context_t  *ac = (cs_xdef_array_context_t *)def->context;

      /* Test if location has at least the pattern of the reference support */
      if (cs_flag_test(ac->loc, cs_flag_dual_face_byc)) {

        /* Sanity check */
        assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_PE));

        const cs_real_t  *flux_array =
          ac->values + cs_cdo_connect->c2e->idx[cm->c_id];
        for (short int e = 0; e < cm->n_ec; e++)
          fluxes[e] = flux_array[e];

      }
      else if (cs_flag_test(ac->loc, cs_flag_primal_face)) {

        const cs_real_t  *i_mflx = ac->values;
        const cs_real_t  *b_mflx = ac->values + cm->bface_shift;

        /* Sanity check */
        assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_PFQ | CS_FLAG_COMP_DEQ |
                             CS_FLAG_COMP_DFQ | CS_FLAG_COMP_PEQ));

        cs_real_t  cell_mflx[3] = {0., 0., 0.};
        cs_reco_cw_cell_vect_from_face_dofs(cm, i_mflx, b_mflx, cell_mflx);

        /* Loop on cell edges */
        for (short int e = 0; e < cm->n_ec; e++)
          fluxes[e] = cm->dface[e].meas * _dp3(cell_mflx, cm->dface[e].unitv);

      }
      else if (cs_flag_test(ac->loc, cs_flag_primal_cell)) {

        /* Retrieve the advection field:
           Switch to a cs_nvec3_t representation */
        cs_nvec3_t  nv;
        cs_nvec3(ac->values + 3*cm->c_id, &nv);

        /* Sanity check */
        assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_DFQ | CS_FLAG_COMP_PEQ));

        /* Loop on cell edges */
        for (short int e = 0; e < cm->n_ec; e++)
          fluxes[e] = nv.meas*cm->dface[e].meas*_dp3(nv.unitv,
                                                     cm->dface[e].unitv);

      }
      else
        bft_error(__FILE__, __LINE__, 0,
                  " Invalid support for evaluating the advection field %s"
                  " at the cell center of cell %ld.", adv->name,
                  (long)cm->c_id);
    }
    break;

  case CS_XDEF_BY_FIELD:
    {
      cs_field_t  *f = (cs_field_t *)def->context;
      const cs_mesh_location_type_t  loc_type =
        cs_mesh_location_get_type(f->location_id);

      switch(loc_type) {

      case CS_MESH_LOCATION_CELLS:
        {
          /* Retrieve the advection field:
             Switch to a cs_nvec3_t representation */
          cs_nvec3_t  nv;
          cs_nvec3(f->val + 3*cm->c_id, &nv);

          /* Sanity check */
          assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_DFQ | CS_FLAG_COMP_PEQ));

          /* Loop on cell edges */
          for (short int e = 0; e < cm->n_ec; e++)
            fluxes[e] = nv.meas*cm->dface[e].meas*_dp3(nv.unitv,
                                                       cm->dface[e].unitv);
        }
        break;

      case CS_MESH_LOCATION_INTERIOR_FACES:
        {
          assert(adv->bdy_field_id > -1);
          cs_field_t  *b_mflx_fld = cs_field_by_id(adv->bdy_field_id);
          assert(b_mflx_fld != NULL);

          const cs_real_t  *b_mflx = b_mflx_fld->val;
          const cs_real_t  *i_mflx = f->val;

          /* Sanity check */
          assert(cs_eflag_test(cm->flag, CS_FLAG_COMP_PFQ | CS_FLAG_COMP_DEQ |
                               CS_FLAG_COMP_DFQ | CS_FLAG_COMP_PEQ));

          cs_real_t  cell_mflx[3] = {0., 0., 0.};
          cs_reco_cw_cell_vect_from_face_dofs(cm, i_mflx, b_mflx, cell_mflx);

          /* Loop on cell edges */
          for (short int e = 0; e < cm->n_ec; e++)
            fluxes[e] = cm->dface[e].meas * _dp3(cell_mflx, cm->dface[e].unitv);
        }
        break;

      default:
        bft_error(__FILE__, __LINE__, 0, "%s: Invalid location type.\nTODO.",
                  __func__);

      }

    }
    break;

  default:
    bft_error(__FILE__, __LINE__, 0, "%s: Incompatible type of definition.",
              __func__);
    break;

  } /* def_type */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief   For each cs_adv_field_t structures, update the values of the
 *          related field(s)
 *
 * \param[in]  t_eval     physical time at which one evaluates the term
 * \param[in]  cur2prev   true or false
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_field_update(cs_real_t    t_eval,
                          bool         cur2prev)
{
  for (int i = 0; i < _n_adv_fields; i++) {

    cs_adv_field_t  *adv = _adv_fields[i];

    /* Sanity checks */
    assert(adv != NULL);

    if (t_eval > 0 && (adv->status & CS_ADVECTION_FIELD_STEADY))
      continue;

    /* GWF and NAVSTO categories of advection fields are updated elsewhere
       except if there is a field defined at vertices */

    if ((adv->status & CS_ADVECTION_FIELD_USER) ||
        (adv->status & CS_ADVECTION_FIELD_LEGACY_FV)) {

      /* Field stored at cell centers */
      assert(adv->cell_field_id > -1);
      cs_field_t  *cfld = cs_field_by_id(adv->cell_field_id);

      /* Copy current field values to previous values */
      if (cur2prev)
        cs_field_current_to_previous(cfld);

      /* Set the new values */
      cs_advection_field_in_cells(adv, t_eval, cfld->val);

      /* Set the new values (only in the case of a user-defined advection
         field */
      if ((adv->status & CS_ADVECTION_FIELD_USER) && adv->bdy_field_id > -1) {

        /* Field storing the boundary normal flux */
        cs_field_t  *bfld = cs_field_by_id(adv->bdy_field_id);

        /* Copy current field values to previous values */
        if (cur2prev)
          cs_field_current_to_previous(bfld);

        cs_advection_field_across_boundary(adv, t_eval, bfld->val);

      }

    }

    if (adv->vtx_field_id > -1) { /* Field stored at vertices */

      cs_field_t  *vfld = cs_field_by_id(adv->vtx_field_id);

      /* Copy current field values to previous values */
      if (cur2prev)
        cs_field_current_to_previous(vfld);

      /* Set the new values */
      cs_advection_field_at_vertices(adv, t_eval, vfld->val);

    }

  } /* Loop on advection fields */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief   Compute the Peclet number in each cell
 *
 * \param[in]      adv        pointer to the advection field struct.
 * \param[in]      diff       pointer to the diffusion property struct.
 * \param[in]      t_eval     time at which one evaluates the advection field
 * \param[in, out] peclet     pointer to an array storing the Peclet number
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_get_peclet(const cs_adv_field_t     *adv,
                        const cs_property_t      *diff,
                        cs_real_t                 t_eval,
                        cs_real_t                 peclet[])
{
  /* Sanity checks */
  assert(adv != NULL);
  assert(diff != NULL);
  assert(peclet != NULL);

  cs_real_t  ptymat[3][3];
  cs_real_3_t  ptydir;
  cs_nvec3_t  adv_c;

  const bool  pty_uniform = cs_property_is_uniform(diff);
  const cs_cdo_quantities_t  *cdoq = cs_cdo_quant;

  /* Get the value of the material property at the first cell center */
  if (pty_uniform)
    cs_property_get_cell_tensor(0, t_eval, diff, false, ptymat);

  for (cs_lnum_t c_id = 0; c_id < cdoq->n_cells; c_id++) {

    /* Get the value of the material property at the cell center */
    if (!pty_uniform)
      cs_property_get_cell_tensor(c_id, t_eval, diff, false, ptymat);

    const cs_real_t  hc = cbrt(cdoq->cell_vol[c_id]);

    cs_advection_field_get_cell_vector(c_id, adv, &adv_c);
    cs_math_33_3_product((const cs_real_t (*)[3])ptymat, adv_c.unitv, ptydir);
    peclet[c_id] = hc * adv_c.meas / _dp3(adv_c.unitv, ptydir);

  }  /* Loop on cells */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief   Compute the Courant number in each cell
 *
 * \param[in]      adv        pointer to the advection field struct.
 * \param[in]      dt_cur     current time step
 * \param[in, out] courant    pointer to an array storing the Courant number
 */
/*----------------------------------------------------------------------------*/

void
cs_advection_get_courant(const cs_adv_field_t     *adv,
                         cs_real_t                 dt_cur,
                         cs_real_t                 courant[])
{
  assert(courant != NULL);  /* Sanity check */
  const cs_cdo_quantities_t  *cdoq = cs_cdo_quant;
  const cs_adjacency_t  *c2f = cs_cdo_connect->c2f;

  assert(adv->cell_field_id > -1); /* field should be defined at cell centers */

  const cs_field_t  *fld = cs_field_by_id(adv->cell_field_id);

# pragma omp parallel for if (cdoq->n_cells > CS_THR_MIN)
  for (cs_lnum_t c_id = 0; c_id < cdoq->n_cells; c_id++) {

    const cs_real_t  *vel_c = fld->val + 3*c_id;
    const cs_real_t  ovol_c = 1./cdoq->cell_vol[c_id];

    cs_real_t  _courant = 0.;
    for (cs_lnum_t  i = c2f->idx[c_id]; i < c2f->idx[c_id+1]; i++) {

      const cs_real_t  *f_area = cs_quant_get_face_vector_area(c2f->ids[i],
                                                               cdoq);
      _courant = fmax(_courant, fabs(_dp3(f_area, vel_c)) * ovol_c);

    }
    courant[c_id] = _courant * dt_cur;

  } /* Loop on cells */

}

/*----------------------------------------------------------------------------*/
/*!
 * \brief   Compute the divergence of the advection field at vertices
 *          Useful for CDO Vertex-based schemes
 *
 * \param[in]      adv         pointer to the advection field struct.
 * \param[in]      t_eval      time at which one evaluates the advection field
 *
 * \return a pointer to an array storing the result
 */
/*----------------------------------------------------------------------------*/

cs_real_t *
cs_advection_field_divergence_at_vertices(const cs_adv_field_t     *adv,
                                          cs_real_t                 t_eval)
{
  CS_UNUSED(t_eval); /* Useful in case of analytic definition */

  cs_real_t  *divergence = NULL;

  if (adv == NULL)
    return divergence;

  const cs_cdo_quantities_t  *cdoq = cs_cdo_quant;
  const cs_cdo_connect_t  *connect = cs_cdo_connect;
  const cs_adjacency_t  *f2e = connect->f2e;
  const cs_adjacency_t  *e2v = connect->e2v;

  BFT_MALLOC(divergence, cdoq->n_vertices, cs_real_t);
  memset(divergence, 0, sizeof(cs_real_t)*cdoq->n_vertices);

  { /* Volume part */
    const cs_xdef_t  *def = adv->definition;

    switch (def->type) {

    case CS_XDEF_BY_ARRAY:
      {
        cs_xdef_array_context_t  *ac = (cs_xdef_array_context_t *)def->context;

        if (cs_flag_test(ac->loc, cs_flag_dual_face_byc)) {

          const cs_adjacency_t  *c2e = connect->c2e;

          for (cs_lnum_t  c_id = 0; c_id < cdoq->n_cells; c_id++) {
            for (cs_lnum_t j = c2e->idx[c_id]; j < c2e->idx[c_id+1]; j++) {

              const cs_lnum_t  e_id = c2e->ids[j];
              const cs_real_t  flx = ac->values[j];
              const cs_lnum_t  eshift = 2*e_id;
              const cs_lnum_t  v0 = e2v->ids[eshift];
              const cs_lnum_t  v1 = e2v->ids[eshift+1];
              const short int  sgn = e2v->sgn[eshift]; /* Sign for v0 */

              divergence[v0] += -sgn*flx;
              divergence[v1] +=  sgn*flx;

            } /* Loop on cell edges */
          } /* Loop on cells */

        }
        else
          bft_error(__FILE__, __LINE__, 0,
                    " %s: Invalid location for the array.", __func__);

      }
      break;

    default:
      bft_error(__FILE__, __LINE__, 0, " %s: Invalid case.", __func__);

    } /* End of switch */

  } /* Volume part */

  /* Boundary part */
  if (adv->n_bdy_flux_defs > 0) {

    for (int def_id = 0; def_id < adv->n_bdy_flux_defs; def_id++) {

      const cs_xdef_t  *def = adv->bdy_flux_defs[def_id];
      const cs_zone_t  *z = cs_boundary_zone_by_id(def->z_id);
      assert(def->support == CS_XDEF_SUPPORT_BOUNDARY);

      switch (def->type) {

      case CS_XDEF_BY_VALUE:
        {
          const cs_real_t  *constant_val = (cs_real_t *)def->context;

          for (cs_lnum_t id = 0; id < z->n_elts; id++) {

            const cs_lnum_t  bf_id = (z->elt_ids == NULL) ? id: z->elt_ids[id];

            _fill_uniform_boundary_flux(cdoq, f2e, e2v, bf_id, constant_val[0],
                                        divergence);

          } /* Loop on boundary faces */
        }
        break; /* by value */

      case CS_XDEF_BY_ARRAY:
        {
          const cs_xdef_array_context_t  *ac =
            (cs_xdef_array_context_t *)def->context;
          const cs_real_t  *val = ac->values;

          assert(ac->stride == 1);
          assert(z->id == 0);

          if (cs_flag_test(ac->loc, cs_flag_primal_face)) {

            for (cs_lnum_t bf_id = 0; bf_id < cdoq->n_b_faces; bf_id++)
              _fill_uniform_boundary_flux(cdoq, f2e, e2v, bf_id, val[bf_id],
                                          divergence);

          }
          else if (cs_flag_test(ac->loc, cs_flag_dual_closure_byf)) {

            const cs_adjacency_t  *bf2v = connect->bf2v;
            const cs_lnum_t  *idx = bf2v->idx;
            assert(idx == ac->index);

            for (cs_lnum_t bf_id = 0; bf_id < cdoq->n_b_faces; bf_id++) {
              for (cs_lnum_t i = idx[bf_id]; i < idx[bf_id+1]; i++) {
                const cs_lnum_t  v_id = bf2v->ids[i];
                divergence[v_id] += val[i];
              }
            }

          }
          else
            bft_error(__FILE__, __LINE__, 0, " %s: Invalid case.", __func__);

        }
        break; /* by_array */

      default:
        bft_error(__FILE__, __LINE__, 0, " %s: Invalid case", __func__);
        break;

      } /* End of switch on the type of definition */
    }   /* Loop on definitions */

  }
  else {

    /* Handle the boundary flux */
    const cs_field_t  *bflx =
      cs_advection_field_get_field(adv, CS_MESH_LOCATION_BOUNDARY_FACES);

    for (cs_lnum_t bf_id = 0; bf_id < cdoq->n_b_faces; bf_id++) {

      const cs_real_t  face_flx = bflx->val[bf_id];
      const cs_real_t  invsurf = 1./cdoq->b_face_surf[bf_id];
      const cs_lnum_t  f_id = cdoq->n_i_faces + bf_id;

      for (cs_lnum_t i = f2e->idx[f_id]; i < f2e->idx[f_id+1]; i++) {

        const cs_lnum_t  e_id = f2e->ids[i];
        const cs_lnum_t  eshift = 2*e_id;
        const cs_lnum_t  v0 = e2v->ids[eshift];
        const cs_lnum_t  v1 = e2v->ids[eshift+1];
        const double  tef = cs_math_surftri(cdoq->vtx_coord + 3*v0,
                                            cdoq->vtx_coord + 3*v1,
                                            cdoq->b_face_center + 3*bf_id);

        const double  weighted_flux = 0.5 * tef * invsurf * face_flx;

        divergence[v0] += weighted_flux;
        divergence[v1] += weighted_flux;

      } /* Loop on face edges */

    } /* Loop on boundary faces */

  } /* Boundary part */

#if defined(HAVE_MPI) /* Synchronization if needed */
  if (connect->interfaces[CS_CDO_CONNECT_VTX_SCAL] != NULL)
    cs_interface_set_sum(connect->interfaces[CS_CDO_CONNECT_VTX_SCAL],
                         cdoq->n_vertices,
                         1,             /* stride */
                         false,         /* interlace (not useful here) */
                         CS_REAL_TYPE,
                         divergence);
#endif  /* MPI */

  return divergence;
}

/*----------------------------------------------------------------------------*/

#undef _dp3

END_C_DECLS
