Introduction

User integration of the SDE for the user-defined variables.

The variables are constant by default. The SDE must be of the form:

$\frac{dT}{dt}=\frac{T - PIP}{Tca}$

T: particle attribute representing the variable

Tca: characteristic time for the sde to be prescribed in the array auxl1

PIP: coefficient of the SDE (pseudo RHS) to be prescribed in the array auxl2.

If the chosen scheme is first order (nordre=1) then, at the first and only call pip is expressed as a function of the quantities of the previous time step (contained in the particle data).

If the chosen scheme is second order (nordre=2) then, at the first call (nor=1) pip is expressed as a function of the quantities of the previous time step, and at the second passage (nor=2) pip is expressed as a function of the quantities of the current time step.

Example

Example of the integration of the SDE (Stochastic Differential Equation).

void
cs_user_lagr_sde(const cs_real_t  dt[],
                 cs_real_t        taup[],
                 cs_real_3_t      tlag[],
                 cs_real_t        tempct[])
{
  /* Initializations
     --------------- */
  cs_lagr_particle_set_t  *p_set = cs_lagr_get_particle_set();
  const cs_lagr_attribute_map_t *p_am = p_set->p_am;
  cs_real_t *tcarac, *pip;
  BFT_MALLOC(tcarac, p_set->n_particles, cs_real_t);
  BFT_MALLOC(pip   , p_set->n_particles, cs_real_t);
  /* Characteristic time of the current SDE
     -------------------------------------- */
  /* Loop on the additional variables */
  for (int i = 0;
       i < cs_glob_lagr_model->n_user_variables;
       i++) {
    for (cs_lnum_t npt = 0; npt < p_set->n_particles; npt++) {
      unsigned char *part = p_set->p_buffer + p_am->extents * npt;
      cs_real_t *usr_var
        = cs_lagr_particle_attr_n(part, p_am, 0, CS_LAGR_USER);
      cs_real_t *prev_usr_var
        = cs_lagr_particle_attr_n(part, p_am, 1, CS_LAGR_USER);
      /* Characteristic time tca of the differential equation,
         This example must be adapted to the case */
      tcarac[npt] = 1.0;
      /* Prediction at the first substep;
         This example must be adapted to the case */
      if (cs_glob_lagr_time_step->nor == 1)
        pip[npt] = prev_usr_var[i];
      /* Correction at the second substep;
         This example must be adapted to the case */
      else
        pip[npt] = usr_var[i];
    }
    /* Integration of the variable ipl
       ------------------------------- */
    cs_lagr_sde_attr(CS_LAGR_USER, tcarac, pip);
  }
  BFT_FREE(tcarac);
  BFT_FREE(pip);
}