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A large-scale view of the code

Our first two courses were about the general notion of object-oriented programming and the implementation of these notions in some typical Gerris classes. Now we turn to the heart of the matter: how does Gerris solve the Navier-Stokes equations? For this we first look at the large scale view, encompassing the basic operations of the code, initialization, velocity advection and tracer advection-diffusion. We then turn to the detailed analysis of tracer advection and diffusion.

Equation and numerical Scheme

The present section should is brief guide of how Gerris treats the advection-diffusion equation of a tracer, [itex]\alpha<\math>,

[itex] \frac{\partial \alpha}{\partial t} + \nabla \cdot ( \mathbf{u} \alpha) = \nabla \cdot ( D \nabla \alpha)+S_\alpha + \nabla \cdot \mathbf{s}_\alpha <\math>

where [itex]\mathbf{u}<\math> is the fluid velocity and [itex]D<\math> is the diffusion coefficient. We have added for generality two types of sources: [itex]S_\alpha<\math> stands for typical sources included, while [itex]\mathbf{s}_\alpha<\math> would be for generic source vector flux.

Gerris adopts a scheme of the form,

[itex] \frac{\alpha^{n+1} -\alpha^{n}}{\Delta t} + \nabla \cdot ( \mathbf{u}^{n+1/2} \alpha^{n+1/2}) = (1-\beta) \nabla \cdot ( D^n \nabla \alpha^n)+\beta \nabla \cdot ( D^{n+1} \nabla \alpha^{n+1})+S_\alpha^n + \nabla \cdot \mathbf{s}^n_\alpha <\math>

where [itex]\beta<\math> is a factor that has to be between 0.5 and 1.0: [itex]\beta=<\math>0.5 (the default value) corresponds to the Crank-Nicholson scheme which is second order accurate in time; [itex]\beta=<\math>1.0 corresponds to the Euler Forward scheme which although is first order accurate in time is more robust against oscillations. The equation is latterly reordered in a form close to a Poisson's equation,

[itex] \alpha^{n+1} - \nabla \cdot ( \beta \: \Delta t \: D^{n+1} \nabla \alpha^{n+1}) = \alpha^{n} - \nabla \cdot (\Delta t \: \mathbf{u}^{n+1/2} \alpha^{n+1/2}) + \nabla \cdot ( (1-\beta) \: \Delta t \: D^n \nabla \alpha^n)+S_\alpha^n <\math>

Adopting the cell as the control volume, [itex]d \Omega<\math>, assuming constant value of the variable [itex]\alpha<\math> in the cell and integrating the above discrete equation over that control volume, it is obtained the following,

• For the temporal term:

[itex] \iiint_\Omega \alpha \: d \Omega = a_v \: \alpha \: \: d \ell_1 d \ell_2 d \ell_3 \, = a_v \: \alpha \: g_c \, h^3 . <\math> where [itex]h<\math> is the size of the cell. For generality we have used a general system of orthogonal curvilinear coordinates given by [itex](x_1,x_2,x_3)<\math>. If [itex]gm_i<\math> are the metric factor of each coordinate [itex]x_i<\math> ([itex]d \ell_i=gm_i d x_i<\math>), we collect them in a single metric factor [itex]g= gm_1 gm_2 gm_3<\math>. The subscript [itex]c<\math> denote that the metric factor is to be computed at the center of mass of the cell. [itex]a_v<\math> is the volume fraction (it is always 1. except in the case that the cell is cut by a solid segment, i.e a mixed cell).

To this term it is applied additionally the divergence theorem in order to calculate it by means of the advection fluxes through the cell's contour,

[itex] \begin{split} \iiint_\Omega \nabla \cdot ( \mathbf{u}^{n+1/2} \alpha^{n+1/2}) \: \Delta t \: d \Omega &= \iint_\Sigma \Delta t ( \mathbf{u}^{n+1/2} \alpha^{n+1/2}) \cdot \mathbf{n} \: d \Sigma \\ &= \sum_{Contour} {u_f}^{n+1/2} \alpha_f^{n+1/2} \Delta t \, s_f g_f h^2 \end{split} <\math>

where [itex]{u_f}<\math> and [itex]{\alpha_f}<\math> are the normal velocity to the face (or solid segment) and the value of the tracer at that face (or solid segment), respectively; [itex]s_f<\math> is the face fraction (it is always 1. except in the case that the face were cut by a solid segment). [itex]g_f<\math> is, again, a metric factor denoting the subscript [itex]f<\math> that is evaluated at the face.

• For the diffusion terms:

These terms are treated similarly to the advection term resulting in, [itex] \begin{split} \iiint_\Omega \nabla \cdot (\Delta t\: D \nabla \alpha) \: d \Omega &= \iint_\Sigma \Delta t\: D \nabla \alpha \cdot \mathbf{n} \: d \Sigma \\ &= \sum_{Contour} \Delta t\: D_f g_f s_f h (h \nabla^f \alpha) \end{split} <\math> \end{itemize}

Reordering the equation, it results finally in the following expression,

[itex] \begin{split}

\alpha^{n+1} &- \frac{1}{a_v g_c h^2}\sum_{contour} \frac{\Delta t \: D_f \,g_f \,  s_f  \, \beta}{gm_f} \left(h \nabla^f \alpha^{n+1} \right)   =\\  \alpha^{n} &- \frac{1}{a_v g_c}  \sum_{contour} \frac{\Delta t  \, s_f g_f}{h}  {u_f}^{n+1/2} \alpha_f^{n+1/2}  +  S_\alpha^n \, \Delta t + \frac{1}{a_v g_c} \sum_{Contour} s_f^n \Delta t \, s_f g_f/h \right \\  &+ \frac{1}{a_v g_c h^2} \sum_{contour} \frac{\Delta t \: D_f  g_f \, s_f \, (1-\beta)}{gm_f} \left(h \nabla^f \alpha^{n} \right)_f
\end{split}


<\math>

where we have take into account, by means of [itex]gm_f<\math>, that a orthogonal curvilinear coordinates system could affect to the expression of the normal gradient.

simulation_run

Let us assume that you are familiar with Stephane's Journal of Computational Physics article and you want to see how it is implemented in code. One way to do it would be to see what is executed line by line using a tool such as ddd. In this course we lead you directly to the interesting part. We skip the bookkeeping and initialisation stuff that is executed first in gerris. The core of the simulation is in simulation.c, specifically in simulation_run(). Let us look at this function. We start with some more bookkeeping:

static void simulation_run (GfsSimulation * sim){  GfsVariable * p, * pmac, * res = NULL;  GfsDomain * domain;  GSList * i;   domain = GFS_DOMAIN (sim);   p = gfs_variable_from_name (domain->variables, "P");  g_assert (p);  pmac = gfs_variable_from_name (domain->variables, "Pmac");  g_assert (pmac);   gfs_simulation_refine (sim);  gfs_simulation_init (sim);   i = domain->variables;  while (i) {    if (GFS_IS_VARIABLE_RESIDUAL (i->data))      res = i->data;    i = i->next;  }

Then we start the various steps of the integration of a time step:

gfs_simulation_set_timestep (sim);  if (sim->time.i == 0) {    gfs_approximate_projection (domain,				&sim->approx_projection_params,				&sim->advection_params,				p, sim->physical_params.alpha, res);    advance_tracers (domain, sim->advection_params.dt/2.);  }  while (sim->time.t < sim->time.end &&	 sim->time.i < sim->time.iend) {    GfsVariable * g[FTT_DIMENSION];    gdouble tstart = gfs_clock_elapsed (domain->timer);     gts_container_foreach (GTS_CONTAINER (sim->events), (GtsFunc) gfs_event_do, sim);

Inside the time loop

Now we are fully inside the time loop.

gfs_simulation_set_timestep (sim);     gfs_predicted_face_velocities (domain, FTT_DIMENSION, &sim->advection_params);     gfs_variables_swap (p, pmac);    gfs_mac_projection (domain,    			&sim->projection_params,     			&sim->advection_params,			p, sim->physical_params.alpha, g);    gfs_variables_swap (p, pmac);     gts_container_foreach (GTS_CONTAINER (sim->events), (GtsFunc) gfs_event_half_do, sim);     gfs_centered_velocity_advection_diffusion (domain,					       FTT_DIMENSION,					       &sim->advection_params,					       g,					       sim->physical_params.alpha);     if (gfs_has_source_coriolis (domain)) {      gfs_poisson_coefficients (domain, sim->physical_params.alpha);      gfs_correct_normal_velocities (domain, 2, p, g, 0.);      gfs_correct_centered_velocities (domain, 2, g, sim->advection_params.dt);      gfs_source_coriolis_implicit (domain, sim->advection_params.dt);      gfs_correct_normal_velocities (domain, 2, p, g, 0.);      gfs_correct_centered_velocities (domain, 2, g, -sim->advection_params.dt);    }     gfs_domain_cell_traverse (domain,			      FTT_POST_ORDER, FTT_TRAVERSE_NON_LEAFS, -1,			      (FttCellTraverseFunc) gfs_cell_coarse_init, domain);    gfs_simulation_adapt (sim);     gfs_approximate_projection (domain,   				&sim->approx_projection_params,     				&sim->advection_params, p, sim->physical_params.alpha, res);     advance_tracers (domain, sim->advection_params.dt);     sim->time.t = sim->tnext;    sim->time.i++;     gts_range_add_value (&domain->timestep, gfs_clock_elapsed (domain->timer) - tstart);    gts_range_update (&domain->timestep);    gts_range_add_value (&domain->size, gfs_domain_size (domain, FTT_TRAVERSE_LEAFS, -1));    gts_range_update (&domain->size);  }  gts_container_foreach (GTS_CONTAINER (sim->events), (GtsFunc) gfs_event_do, sim);    gts_container_foreach (GTS_CONTAINER (sim->events),			 (GtsFunc) gts_object_destroy, NULL);}

From now on it becomes difficult to offer a traditional written description of the course. Instead, in a class the teacher should open a text editor, projecting it on screen in front of the students, and navigate through the code. For this we shall use an interesting feature of the text editors.

Tip: Navigate using the TAGS file

In the above example, we have seen that we can gain some information about the macros, classes, structures and functions of the code by navigating in the code from one definition to the next. This navigation is not always immediately enlightening: we must resort to it only when we have enough general understanding of the code. To perform this navigation in a more efficient way than grep, we can use the following trick:

• Create the TAGS file :
% cd src
% make tags


You can then use the tags in emacs, nedit or vim.

1. In emacs:
1. open any *.c or *.h file with emacs.
2. Position the cursor on a function or variable.
3. do ESC . to find its definition. (or M-. using the emacs Meta key convention )
2. In vim, nedit ... : vim, nedit users please contribute ...

You can now try to use this trick on your own editor. It is a good idea to try the following: find any place in the code where the GFS_VALUE macro is used — that should be easy — and then click on it. Type M-. in emacs — or the equivalent in vim — and get to the definition, then repeat the operation to display the declarations and definitions above.

From now on, the reader or listener to the course should open his text editor and navigate freely in the code, trying to find the implementation of the numerical methods. We guide the reader below for one particular scheme: the tracer advection.

Basic scheme, structures and modules for tracer advection

The Bell-Collela-Glaz scheme and its adaptation

Gerris implements a variant of Godunov's scheme for advection that is second order in time and space. The relevant reference is :

(BCG) J. B. Bell, P. Colella, H. M. Glaz - A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys. 85, 257-283, (1989).

The BCG scheme is further simplified following :

D. F. Martin, An adaptive cell-centered pro jection method for the incompressible Euler equations, Ph.D. thesis, University of California, Berkeley (1998).

D. F. Martin, P. Colella, A cell-centered adaptive projection method for the incompressible euler equations, J. Comput. Phys. 163, 271-312 (2000).

The advection scheme is also discussed in Stephane's Journal of Computational Physics article, Section 5, page 21.

To fully understand the code, you need to be familiar with the relevant parts of the above references. In these course notes we will not reproduce the corresponding developments, as this is a course about programming and not numerical schemes, but the live course sessions should include a short presentation of the version of the BCG scheme used by Gerris.

The tracer is updated though the function advance_tracers. Click on it, do M-. to navigate and observe the function:

static void advance_tracers (GfsDomain * domain, gdouble dt){  GSList * i = domain->variables;  while (i) {    if (GFS_IS_VARIABLE_TRACER_VOF (i->data)) {      GfsVariableTracer * t = i->data;       t->advection.dt = dt;      gfs_tracer_vof_advection (domain, &t->advection);      gfs_domain_variable_centered_sources (domain, i->data, i->data, t->advection.dt);    }    else if (GFS_IS_VARIABLE_TRACER (i->data)) {      GfsVariableTracer * t = i->data;       t->advection.dt = dt;      gfs_tracer_advection_diffusion (domain, &t->advection);      gfs_domain_cell_traverse (domain,				FTT_POST_ORDER, FTT_TRAVERSE_NON_LEAFS, -1,				(FttCellTraverseFunc) GFS_VARIABLE1 (t)->fine_coarse, t);    }    i = i->next;  }  }

We see that we have two sub-functions gfs_tracer_vof_advection and gfs_tracer_advection_diffusion. One uses the VOF scheme to advect non-diffusive interfaces. The second advects traditional tracers such as temperature. To understand the advection scheme we need to look at several important structures.

GfsVariableTracer object

This is an object class similar to those we have seen before. The full header for that class is:

/* GfsVariableTracer: header */ typedef struct _GfsVariableTracer                GfsVariableTracer; struct _GfsVariableTracer {  /*< private >*/  GfsVariable parent;   /*< public >*/  GfsAdvectionParams advection;}; #define GFS_VARIABLE_TRACER(obj)            GTS_OBJECT_CAST (obj,\					           GfsVariableTracer,\					           gfs_variable_tracer_class ())#define GFS_IS_VARIABLE_TRACER(obj)         (gts_object_is_from_class (obj,\					     gfs_variable_tracer_class ())) GfsVariableClass * gfs_variable_tracer_class  (void);

As we shall see below, however, we do not need the full header to understand the functionality of this object. All we need to know is

struct _GfsVariableTracer {  /*< private >*/  GfsVariable parent;   /*< public >*/  GfsAdvectionParams advection;};

and even more economically, all we need is to know about the member advection.

 GfsAdvectionParams advection;


So we turn to the defintion of its type:

struct _GfsAdvectionParams {  gdouble cfl, dt;                      /* self-explanatory */  GfsVariable * v, * fv, ** u, ** g;    /* v:  typically, the GfsVariable that we are advecting. 					   fv: typically the flux of the GfsVariable. 					   u[c] is the component of velocity in direction c,					           c goes form 0 to 2 in 3D, from 0 to 1 in 2D .					   g[c] is the component of a gradient variable.					*/  GfsCenterGradient gradient;           /* A function pointer, so that the function					   par->gradient(cell, c, v->i) should return					   the value of the c component of the gradient 					   of variable v at the center of the cell. */   gboolean use_centered_velocity;       /* self-explanatory */  GfsUpwinding upwinding;               /* self-explanatory */  GfsFaceAdvectionFluxFunc flux;        /* A function pointer, that is used to compute fluxes */  GfsAdvectionScheme scheme;            /* The scheme used */  gboolean average;                     /* Whether to use averaging in gfs_advection_update() */};

(The comments are not in the source code, they were added for this course.) First we notice that this structure is not a Gerris Object: it does not have a parent, or a related class, and is just a C structure. However it has an initialisation function, and of course it carries a lot of information. Vectors are of the form  ** u, so that  u[c] is a pointer to the c component of the vector u.

We concentrate for now on the second type of advection and, clicking on the corresponding function we see:

/** * gfs_tracer_advection_diffusion: * @domain: a #GfsDomain. * @par: the advection parameters. * * Advects the @v field of @par using the current face-centered (MAC) * velocity field. */void gfs_tracer_advection_diffusion (GfsDomain * domain,				     GfsAdvectionParams * par){  GfsSourceDiffusion * d;   g_return_if_fail (domain != NULL);  g_return_if_fail (par != NULL);   gfs_domain_timer_start (domain, "tracer_advection_diffusion");   if ((d = source_diffusion (par->v))) {    GfsVariable * rhs;     rhs = gfs_temporary_variable (domain);    gfs_domain_cell_traverse (domain, FTT_PRE_ORDER, FTT_TRAVERSE_LEAFS, -1,			      (FttCellTraverseFunc) gfs_cell_reset, rhs);    variable_sources (domain, par, rhs, NULL);    variable_diffusion (domain, d, par, rhs, NULL);    gts_object_destroy (GTS_OBJECT (rhs));  }  else {    variable_sources (domain, par, par->v, NULL);    gfs_domain_bc (domain, FTT_TRAVERSE_LEAFS, -1, par->v);  }   gfs_domain_timer_stop (domain, "tracer_advection_diffusion");}

At this stage the reader may be a little lost: where is the function that advects the tracer? We do see the lines

gfs_domain_cell_traverse (domain, FTT_PRE_ORDER, FTT_TRAVERSE_LEAFS, -1,			      (FttCellTraverseFunc) gfs_cell_reset, rhs);

It involves the important function gfs_domain_cell_traverse. We explain it in the next subsection. The function variable_sources is the one that advects the tracer, we explain it in the next section.

gfs_domain_cell_traverse() module

This is an important function which is very frequently used in the code. It allows to perform a sequence of calls of a given function, say func, one call for each cell of the tree. It is used to perform operations local to each cell. In a sense it is the equivalent of a do loop over an NxN array in a classical fortran or C code. The arguments of the function are (cell, data) where cell is the current cell in the tree, and data is the data passed as last argument of gfs_domain_cell_traverse(). In the example above, the function func is  gfs_cell_reset  and the data is rhs.

We can understand this function better if have a look at the Reference Manual . Click on the link to the description of gfs_domain_cell_traverse.

Following the links there we can understand a lot about the function. Let us summarize the most important facts. When FTT_PRE_ORDER is used the function is applied first to the root and then to the children. It progressively descends until max_depth, or the leaves, are visited. When FTT_POST_ORDER is used the function is applied first to the leaves and then to the children. The flag FTT_TRAVERSE_LEAFS  is pretty explicit. Another possible value for the flag would be FTT_TRAVERSE_NON_LEAFS. Notice that when the flag is set to FTT_TRAVERSE_LEAFS  the argument @order has no effect. Notice also that we do not give explicit values to the various order and flag arguments, instead we use values defined in macros, which makes them more explicit.

The module gfs_domain_cell_traverse is a very useful and widely used function. However in the example above the call to gfs_cell_reset does not do much: it justs resets rhs to zero. We must find the actual advection in another function at a deeper level.

Note

As usual it is important to separate functionality from implementation. It is enough to understand how the Gerris code operate to learn about functionality. If we want to know more about the gfs_domain_cell_traverse function, we may feel the urge to look at its source code. This may not be a good idea, especially at this stage of the course. Anyway, should we embark on it, it requires, as usual, a lot of navigation. We explore in a sequence the functions:

1. gfs_domain_cell_traverse
2. gts_container_foreach
3. static void box_traverse
4. void ftt_cell_traverse
5. cell_traverse_pre_order_all

(Navigate on your screen through the functions.)

variable_sources() module

Having a look at this function (we do not display it here, but you should look at it now using you text editor) reveals a large number of functions. The gfs_domain_cell_traverse function is called many times to apply various functions. These functions are

• gfs_cell_reset
• par->flux (a fonction pointer, which will be elucidated below to be gfs_face_advection_flux)

We detail the various functions below.

gfs_cell_reset

Again the call to gfs_cell_reset does not do much: it justs resets par->fv to zero.

The documentation section above this function says what it does:

/** * gfs_cell_advected_face_values: * @cell: a #FttCell. * @par: the advection parameters. * * Fills the face variable (@v field of #GfsFaceStateVector) of all the * faces of @cell with the advected value of variable @par->v at time * t + dt/2. */void gfs_cell_advected_face_values (FttCell * cell,				    const GfsAdvectionParams * par)

The v field of GfsFaceStateVector is a special field defined on the faces which we will use to store the values of the tracer variable to be advected.

What is the variable par->v ? To find out, we need to look for where the GfsAdvectionParam variable par is initialized. On can search using grep as in the paragraph about par->flux below. Finally one discovers that initialization is done in variable.c as follows:

static void variable_tracer_init (GfsVariableTracer * v){  gfs_advection_params_init (&v->advection);  v->advection.gradient = gfs_center_van_leer_gradient;  v->advection.flux = gfs_face_advection_flux;  v->advection.v = GFS_VARIABLE1 (v);  v->advection.fv = NULL;  GFS_VARIABLE1 (v)->description = g_strdup ("Tracer");}

where the GFS_VARIABLE1 macro is

#define GFS_VARIABLE1(obj)            GTS_OBJECT_CAST (obj,\					         GfsVariable,\					         gfs_variable_class ())

So finally the par->v variable is just the tracer variable we are advecting, which should not be surprising. Now, look at the source of the function. This function appears to be rather complex.

void gfs_cell_advected_face_values (FttCell * cell,				    const GfsAdvectionParams * par){  FttComponent c;  gdouble size;  GfsStateVector * s;   g_return_if_fail (cell != NULL);  g_return_if_fail (par != NULL);   s = GFS_STATE (cell);  size = ftt_cell_size (cell);  for (c = 0; c < FTT_DIMENSION; c++) {    gdouble unorm = par->use_centered_velocity ?      par->dt*GFS_VALUE (cell, par->u[c])/size :      par->dt*(s->f[2*c].un + s->f[2*c + 1].un)/(2.*size);    gdouble g = (* par->gradient) (cell, c, par->v->i);    gdouble vl = GFS_VALUE (cell, par->v) + MIN ((1. - unorm)/2., 0.5)*g;    gdouble vr = GFS_VALUE (cell, par->v) + MAX ((- 1. - unorm)/2., -0.5)*g;    gdouble src = par->dt*gfs_variable_mac_source (par->v, cell)/2.;    gdouble dv; #if FTT_2D    dv = transverse_term (cell, size, FTT_ORTHOGONAL_COMPONENT (c), par);#else  /* FTT_3D */    static FttComponent orthogonal[FTT_DIMENSION][2] = {      {FTT_Y, FTT_Z}, {FTT_X, FTT_Z}, {FTT_X, FTT_Y}    };     dv =  transverse_term (cell, size, orthogonal[c][0], par);    dv += transverse_term (cell, size, orthogonal[c][1], par);#endif /* FTT_3D */     s->f[2*c].v     = vl + src - dv;    s->f[2*c + 1].v = vr + src - dv;  }}
Figure 1 (from Dan Martin's PhD thesis)

It actually implements the BCG scheme. In the code vl and vr are the advected "left" and ""right" value of Figure 1. Look at page 21, Eq. (19) of Stephane's Journal of Computational Physics article for the expression of the "Left State" and "Right State" of the faces (you should really be familiar with Godunov's method to follow this). The gradient g is computed using the gradient function specified in the gradient member of GfsAdvectionParam par . The member is initialized in variable.c in the function

static void variable_tracer_init (GfsVariableTracer * v)


shown above. The gradient is defined there as the gfs_center_van_leer_gradient() module, a self explanatory name.

par->flux

One of the important functions for the implementation of the u grad T term is the function par->flux . This is not a fixed name for a function but a pointer to a function that defines the advection flux. We cannot click on it to determine how it is defined ! Doing a

% grep flux *.c


reveals too many lines to be useful. Doing a

% grep par\-\>flux *.c


seems to reveal the definition of the flux function in the file advection.c in the module gfs_advection_params_read. There are actually several options for this function. (Look at the source code: you can see the various options.) The option can be selected in the simulation file (for instance simulation.gfs). Only one of them is the default one. So where is the default initialized? (Navigate to the initialization function of the object GfsAdvectionParams). Navigation, as you see, does not help. We need to do a

% grep flux *.c  | grep =


which yields two initializations, one in the module simulation_init() of simulation.c:

object->advection_params.flux = gfs_face_velocity_advection_flux;

and another in variable.c :

v->advection.flux = gfs_face_advection_flux;

The first piece of code initializes the function flux for the velocity advection "in" the simulation parameters of the object of type GfsSimulation. This object describes the whole Navier-Stokes resolution procedure, and its AdvectionParam is naturally that for velocity advection in Navier--Stokes. The other piece of code initializes it in the initialisation function of GfsVariableTracer.

If we look in this function we see the comments

/** * gfs_face_advection_flux: * @face: a #FttCellFace. * @par: the advection parameters. * * Adds to variable @par->fv, the value of the (conservative) * advection flux of the face variable through @face. * * This function assumes that the face variable has been previously * defined using gfs_cell_advected_face_values(). */

and then some source code. One part is particularly telling:

flux = GFS_FACE_FRACTION (face)*GFS_FACE_NORMAL_VELOCITY (face)*par->dt*    gfs_face_upwinded_value (face, GFS_FACE_UPWINDING, NULL)/ftt_cell_size (face->cell);

so now we have the hint that the flux has the well-known form

flux = u_face * T_face


The face variable T_face is computed in an upwinded manner following the BCG scheme. It uses the following function:

gfs_face_upwinded_value
/** * gfs_face_upwinded_value: * @face: a #FttCellFace. * @upwinding: type of upwinding. * @u: the cell-centered velocity. * * This function assumes that the face variable has been previously * defined using gfs_cell_advected_face_values(). * * Returns: the upwinded value of the face variable.   */gdouble gfs_face_upwinded_value (const FttCellFace * face,				 GfsUpwinding upwinding,				 GfsVariable ** u)

To see how this function works, we need to understand some of the functions defined in ftt.h and ftt.c. These are part of the Fully Threaded Tree structure explained in the next session.

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