poisson.h

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/** @file poisson.h
 */
/**
# Multigrid Poisson--Helmholtz solvers
 
We want to solve Poisson--Helmholtz equations of the general form
\f[
L(a) = \nabla\cdot (\alpha\nabla a) + \lambda a = b
\f]
This can be done efficiently using a multigrid solver. 
 
An important aspect of Poisson--Helmholtz equations is that the
operator \f$L()\f$ is linear. This property can be used to build better
estimates of a solution by successive *corrections* to an initial
guess. If we define an approximate solution \f$\tilde{a}\f$ as
\f[
\tilde{a} + da = a
\f]
where \f$a\f$ is the exact (unknown) solution, using the linearity of the
operator we find that \f$da\f$ verifies
\f[
L(da) = b - L(\tilde{a})
\f]
where the right-hand-side is often called the *residual* of the
approximate solution \f$\tilde{a}\f$.
 
## Multigrid cycle
 
Here we implement the multigrid cycle proper. Given an initial guess
*a*, a residual *res*, a correction field *da* and a relaxation
function *relax*, we will provide an improved guess at the end of the
cycle. */
 
trace
void mg_cycle (scalar * a, scalar * res, scalar * da,
         void (* relax) (scalar * da, scalar * res, 
             int depth, void * data),
         void * data,
         int nrelax, int minlevel, int maxlevel)
{
 
  /**
  We first define the residual on all levels. */
 
  restriction (res);
 
  /**
  We then proceed from the coarsest grid (*minlevel*) down to the
  finest grid. */
 
  minlevel = min (minlevel, maxlevel);
  for (int l = minlevel; l <= maxlevel; l++) {
 
    /**
    On the coarsest grid, we take zero as initial guess. */
 
    if (l == minlevel)
      for (int _i = 0; _i < l; _i++)
  for (int _s = 0; _s < 1; _s++) /* scalar in da */
    /* foreach_blockf */
      s[] = 0.;
 
    /**
    On all other grids, we take as initial guess the approximate solution
    on the coarser grid bilinearly interpolated onto the current grid. */
 
    else {
      boundary_level (da, l - 1);
      for (int _l = 0; _l < l; _l++)
  for (int _s = 0; _s < 1; _s++) /* scalar in da */
    /* foreach_blockf */
      s[] = bilinear (point, s);
    }
    
    /**
    We then apply homogeneous boundary conditions and do several
    iterations of the relaxation function to refine the initial guess. */
 
    for (int i = 0; i < nrelax; i++) {
      boundary_level (da, l);
      relax (da, res, l, data);
    }
  }
 
  /**
  And finally we apply the resulting correction to *a*. */
 
  for (int _i = 0; _i < _N; _i++) /* foreach */ {
    scalar s, ds;
    for (s, ds in a, da)
      /* foreach_blockf */
  s[] += ds[];
  }
}
 
/**
## Multigrid solver
 
The multigrid solver itself uses successive calls to the multigrid
cycle to refine an initial guess until a specified tolerance is
reached. 
 
The maximum number of iterations is controlled by *NITERMAX* and the
tolerance by *TOLERANCE* with the default values below. */
 
int NITERMAX = 100, NITERMIN = 1;
double TOLERANCE = 1e-3 [*];
 
/**
Information about the convergence of the solver is returned in a structure. */
 
typedef struct {
  int i;              // number of iterations
  double resb, resa;  // maximum residual before and after the iterations
  double sum;         // sum of r.h.s.
  int nrelax;         // number of relaxations
  int minlevel;       // minimum level of the multigrid hierarchy
} mgstats;
 
/**
The user needs to provide a function which computes the residual field
(and returns its maximum) as well as the relaxation function. The
user-defined pointer *data* can be used to pass arguments to these
functions. The optional number of relaxations is *nrelax* and *res* is
an optional list of fields used to store the residuals. The minimum
level of the hierarchy can be set (default is zero i.e. the root
cell). */
 
trace
mgstats mg_solve (scalar * a, scalar * b,
      double (* residual) (scalar * a, scalar * b, scalar * res,
               void * data),
      void (* relax) (scalar * da, scalar * res, int depth, 
          void * data),
      void * data = NULL,
      int nrelax = 4,
      scalar * res = NULL,
      int minlevel = 0,
      double tolerance = TOLERANCE)
{
 
  /**
  We allocate a new correction and residual field for each of the scalars
  in *a*. */
 
  scalar * da = list_clone (a), * pres = res;
  if (!res)
    res = list_clone (b);
 
  /**
  The boundary conditions for the correction fields are the
  *homogeneous* equivalent of the boundary conditions applied to
  *a*. */
 
  for (int b = 0; b < nboundary; b++)
    for (int _s = 0; _s < 1; _s++) /* scalar in da */
      s.boundary[b] = s.boundary_homogeneous[b];
  
  /**
  We initialise the structure storing convergence statistics. */
 
  mgstats s = {0};
  double sum = 0.;
  scalar rhs = b[0];
  foreach (reduction(+:sum))
    sum += rhs[];
  s.sum = sum;
  s.nrelax = nrelax > 0 ? nrelax : 4;
  
  /**
  Here we compute the initial residual field and its maximum. */
 
  double resb;
  resb = s.resb = s.resa = (* residual) (a, b, res, data);
 
  /**
  We then iterate until convergence or until *NITERMAX* is reached. Note
  also that we force the solver to apply at least one cycle, even if the
  initial residual is lower than *TOLERANCE*. */
  
  for (s.i = 0;
       s.i < NITERMAX && (s.i < NITERMIN || s.resa > tolerance);
       s.i++) {
    mg_cycle (a, res, da, relax, data,
        s.nrelax,
        minlevel,
        grid->maxdepth);
    s.resa = (* residual) (a, b, res, data);
 
    /**
    We tune the number of relaxations so that the residual is reduced
    by between 2 and 20 for each cycle. This is particularly useful
    for stiff systems which may require a larger number of relaxations
    on the finest grid. */
 
#if 1
    if (s.resa > tolerance) {
      if (resb/s.resa < 1.2 && s.nrelax < 100)
  s.nrelax++;
      else if (resb/s.resa > 10 && s.nrelax > 2)
  s.nrelax--;
    }
#else
    if (s.resa == resb) /* convergence has stopped!! */
      break;
    if (s.resa > resb/1.1 && p.minlevel < grid->maxdepth)
      p.minlevel++;
#endif
 
    resb = s.resa;
  }
  s.minlevel = minlevel;
  
  /**
  If we have not satisfied the tolerance, we warn the user. */
 
  if (s.resa > tolerance) {
    scalar v = a[0]; // fixme: should not be necessary
    fprintf (ferr, 
       "src/poisson.h:%d: warning: convergence for %s not reached after %d iterations\n"
       "  res: %g sum: %g nrelax: %d tolerance: %g\n", LINENO, v.name,
       s.i, s.resa, s.sum, s.nrelax, tolerance), fflush (ferr);
  }
    
  /**
  We deallocate the residual and correction fields and free the lists. */
 
  if (!pres)
    delete (res), free (res);
  delete (da), free (da);
 
  return s;
}
 
/**
## Application to the Poisson--Helmholtz equation
 
We now apply the generic multigrid solver to the Poisson--Helmholtz equation
\f[
\nabla\cdot (\alpha\nabla a) + \lambda a = b
\f]
We first setup the data structure required to pass the extra
parameters \f$\alpha\f$ and \f$\lambda\f$. We define \f$\alpha\f$ as a face
vector field because we need values at the face locations
corresponding to the face gradients of field \f$a\f$. 
 
*alpha* and *lambda* are declared as *(const)* to indicate that the
function works also when *alpha* and *lambda* are constant vector
(resp. scalar) fields. If *tolerance* is set, it supersedes the
default *TOLERANCE* of the multigrid solver, *nrelax* controls the
initial number of relaxations (default is one), *minlevel* controls
the minimum level of the hierarchy (default is one) and *res* is an
optional list of fields used to store the final residual (which can be
useful to monitor convergence). */
 
struct Poisson {
  scalar a, b;
  const vector alpha;
  const scalar lambda;
  double tolerance;
  int nrelax, minlevel;
  scalar * res;
#if EMBED
  double (* embed_flux) (Point, scalar, vector, double *);
#endif
};
 
/**
We can now write the relaxation function. We first recover the extra
parameters from the data pointer. */
 
static void relax (scalar * al, scalar * bl, int l, void * data)
{
  scalar a = al[0], b = bl[0];
  struct Poisson * p = (struct Poisson *) data;
  const vector alpha = p->alpha;
  const scalar lambda = p->lambda;
 
  /**
  We use either Jacobi (under)relaxation, Gauss-Seidel or we directly
  reuse values as soon as they are updated. For Jacobi, we need to
  allocate space for the new field *c*. Jacobi is useful mostly as it
  gives results which are independent of the order in which the cells
  are traversed. This is not the case for the simple traversal, which
  means for example that results will depend on whether a tree or a
  multigrid is used (because cells will be traversed in a different
  order). The same comment applies to OpenMP or MPI parallelism. In
  practice however Jacobi convergence tends to be slower than simple
  reuse. */
  
#if JACOBI
  scalar c[];
#else
  scalar c = a;
#endif
 
  /**
  On GPUs, we use red/black Gauss-Seidel relaxation, which requires
  two loops (for odd/even indices). Note also that, unlike the other
  option, red/black relaxation should be deterministic. */
  
#if GAUSS_SEIDEL || _GPU
  for (int parity = 0; parity < 2; parity++)
    for (int _i = 0; _i < l, nowarning; _i++)
      if (level == 0 || ((point.i + parity) % 2) != (point.j % 2))
#else
  for (int _i = 0; _i < l, nowarning; _i++)
#endif
  {
 
    /**
    We use the face values of \f$\alpha\f$ to weight the gradients of the
    5-points Laplacian operator. We get the relaxation function. */
 
    double n = - sq(Delta)*b[], d = - lambda[]*sq(Delta);
    for (int _d = 0; _d < dimension; _d++) {
      n += alpha.x[1]*a[1] + alpha.x[]*a[-1];
      d += alpha.x[1] + alpha.x[];
    }
#if EMBED
    if (p->embed_flux) {
      double c, e = p->embed_flux (point, a, alpha, &c);
      n -= c*sq(Delta);
      d += e*sq(Delta);
    }
    if (!d)
      c[] = 0., b[] = 0.;
    else
#endif // EMBED
      c[] = n/d;
  }
 
  /**
  For weighted Jacobi we under-relax with a weight of 2/3. */
  
#if JACOBI
  for (int _i = 0; _i < l; _i++)
    a[] = (a[] + 2.*c[])/3.;
#endif
  
#if TRASH
  scalar a1[];
  for (int _i = 0; _i < l; _i++)
    a1[] = a[];
  /* trash: a */;
  for (int _i = 0; _i < l; _i++)
    a[] = a1[];
#endif
}
 
/**
The equivalent residual function is obtained in a similar way in the
case of a Cartesian grid, however the case of the tree mesh
requires more careful consideration... */
 
static double residual (scalar * al, scalar * bl, scalar * resl, void * data)
{
  scalar a = al[0], b = bl[0], res = resl[0];
  struct Poisson * p = (struct Poisson *) data;
  const vector alpha = p->alpha;
  const scalar lambda = p->lambda;
  double maxres = 0.;
#if TREE
  /* conservative coarse/fine discretisation (2nd order) */
  vector g[];
  for (int _i = 0; _i < _N; _i++) /* foreach_face */
    g.x[] = alpha.x[]*face_gradient_x (a, 0);
  foreach (reduction(max:maxres), nowarning) {
    res[] = b[] - lambda[]*a[];
    for (int _d = 0; _d < dimension; _d++)
      res[] -= (g.x[1] - g.x[])/Delta;
#if EMBED
    if (p->embed_flux) {
      double c, e = p->embed_flux (point, a, alpha, &c);
      res[] += c - e*a[];
    }
#endif // EMBED    
    if (fabs (res[]) > maxres)
      maxres = fabs (res[]);
  }
#else // !TREE
  /* "naive" discretisation (only 1st order on trees) */
  foreach (reduction(max:maxres), nowarning) {
    res[] = b[] - lambda[]*a[];
    for (int _d = 0; _d < dimension; _d++)
      res[] += (alpha.x[0]*face_gradient_x (a, 0) -
    alpha.x[1]*face_gradient_x (a, 1))/Delta;  
#if EMBED
    if (p->embed_flux) {
      double c, e = p->embed_flux (point, a, alpha, &c);
      res[] += c - e*a[];
    }
#endif // EMBED
    if (fabs (res[]) > maxres)
      maxres = fabs (res[]);
  }
#endif // !TREE
  return maxres;
}
 
/**
## User interface
 
Finally we provide a generic user interface for a Poisson--Helmholtz
equation of the form
\f[
\nabla\cdot (\alpha\nabla a) + \lambda a = b
\f] */
 
trace
mgstats poisson (scalar a, scalar b,
     const vector alpha = {{-1}},
     const scalar lambda = {-1},
     double tolerance = 0.,
     int nrelax = 4,
     int minlevel = 0,
     scalar * res = NULL,
     double (* flux) (Point, scalar, vector, double *) = NULL)
{
 
  /**
  If \f$\alpha\f$ or \f$\lambda\f$ are not set, we replace them with constant
  unity vector (resp. zero scalar) fields. Note that the user is free to
  provide \f$\alpha\f$ and \f$\beta\f$ as constant fields. */
 
  if (alpha.x.i < 0)
    alpha[] = {1.,1.,1.};
  if (lambda.i < 0)
    lambda[] = 0.;
 
 
 
  /**
  We need \f$\alpha\f$ and \f$\lambda\f$ on all levels of the grid. */
 
  restriction ({alpha,lambda});
 
  /**
  If *tolerance* is set it supersedes the default of the multigrid
  solver. */
 
  double defaultol = TOLERANCE;
  if (tolerance)
    TOLERANCE = tolerance;
 
  struct Poisson p = {a, b, alpha, lambda, tolerance, nrelax, minlevel, res };
#if EMBED
  if (!flux && a.boundary[embed] != symmetry)
    p.embed_flux = embed_flux;
  else
    p.embed_flux = flux;
#endif // EMBED
  mgstats s = mg_solve ({a}, {b}, residual, relax, &p,
      nrelax, res, max(1, minlevel));
 
  /**
  We restore the default. */
 
  if (tolerance)
    TOLERANCE = defaultol;
 
  return s;
}
 
/**
## Projection of a velocity field
 
The function below "projects" the velocity field *u* onto the space of
divergence-free velocity fields i.e.
\f[
\mathbf{u}_f^{n+1} \leftarrow \mathbf{u}_f - \Delta t\alpha\nabla p
\f]
so that
\f[
\nabla\cdot\mathbf{u}_f^{n+1} = 0
\f]
This gives the Poisson equation for the pressure
\f[
\nabla\cdot(\alpha\nabla p) = \frac{\nabla\cdot\mathbf{u}_f}{\Delta t}
\f] */
 
trace
mgstats project (vector uf, scalar p,
     const vector alpha = unityf,
     double dt = 1.,
     int nrelax = 4)
{
  
  /**
  We allocate a local scalar field and compute the divergence of
  \f$\mathbf{u}_f\f$. The divergence is scaled by *dt* so that the
  pressure has the correct dimension. */
 
  scalar div[];
  for (int _i = 0; _i < _N; _i++) /* foreach */ {
    div[] = 0.;
    for (int _d = 0; _d < dimension; _d++)
      div[] += uf.x[1] - uf.x[];
    div[] /= dt*Delta;
  }
 
  /**
  We solve the Poisson problem. The tolerance (set with *TOLERANCE*) is
  the maximum relative change in volume of a cell (due to the divergence
  of the flow) during one timestep i.e. the non-dimensional quantity 
  \f[
  |\nabla\cdot\mathbf{u}_f|\Delta t 
  \f] 
  Given the scaling of the divergence above, this gives */
 
  mgstats mgp = poisson (p, div, alpha,
       tolerance = TOLERANCE/sq(dt), nrelax = nrelax);
 
  /**
  And compute \f$\mathbf{u}_f^{n+1}\f$ using \f$\mathbf{u}_f\f$ and \f$p\f$. */
 
  for (int _i = 0; _i < _N; _i++) /* foreach_face */
    uf.x[] -= dt*alpha.x[]*face_gradient_x (p, 0);
 
  return mgp;
}