/** # Multigrid Poisson--Helmholtz solvers We want to solve Poisson--Helmholtz equations of the general form $$L(a) = \nabla\cdot (\alpha\nabla a) + \lambda a = b$$ This can be done efficiently using a multigrid solver. An important aspect of Poisson--Helmholtz equations is that the operator $L()$ 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 $\tilde{a}$ as $$\tilde{a} + da = a$$ where $a$ is the exact (unknown) solution, using the linearity of the operator we find that $da$ verifies $$L(da) = b - L(\tilde{a})$$ where the right-hand-side is often called the *residual* of the approximate solution $\tilde{a}$. ## 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. */ 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) foreach_level_or_leaf (l) for (scalar s in da) foreach_blockf (s) 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 foreach_level (l) for (scalar s in da) foreach_blockf (s) s[] = bilinear (point, s); /** We then apply homogeneous boundary conditions and do several iterations of the relaxation function to refine the initial guess. */ boundary_level (da, l); for (int i = 0; i < nrelax; i++) { relax (da, res, l, data); boundary_level (da, l); } } /** And finally we apply the resulting correction to *a*. */ foreach() { scalar s, ds; for (s, ds in a, da) foreach_blockf (s) 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). */ 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 (scalar s 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, "WARNING: convergence for %s not reached after %d iterations\n" " res: %g sum: %g nrelax: %d tolerance: %g\n", 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 $$\nabla\cdot (\alpha\nabla a) + \lambda a = b$$ We first setup the data structure required to pass the extra parameters $\alpha$ and $\lambda$. We define $\alpha$ as a face vector field because we need values at the face locations corresponding to the face gradients of field $a$. *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) face 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) face vector alpha = p->alpha; (const) scalar lambda = p->lambda; /** We use either Jacobi (under)relaxation 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 /** We use the face values of $\alpha$ to weight the gradients of the 5-points Laplacian operator. We get the relaxation function. */ foreach_level_or_leaf (l) { double n = - sq(Delta)*b[], d = - lambda[]*sq(Delta); foreach_dimension() { 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 foreach_level_or_leaf (l) a[] = (a[] + 2.*c[])/3.; #endif #if TRASH scalar a1[]; foreach_level_or_leaf (l) a1[] = a[]; trash ({a}); foreach_level_or_leaf (l) 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) face vector alpha = p->alpha; (const) scalar lambda = p->lambda; double maxres = 0.; #if TREE /* conservative coarse/fine discretisation (2nd order) */ face vector g[]; foreach_face() g.x[] = alpha.x[]*face_gradient_x (a, 0); foreach (reduction(max:maxres), nowarning) { res[] = b[] - lambda[]*a[]; foreach_dimension() 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[]; foreach_dimension() 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 $$\nabla\cdot (\alpha\nabla a) + \lambda a = b$$ */ mgstats poisson (scalar a, scalar b, (const) face 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 $\alpha$ or $\lambda$ are not set, we replace them with constant unity vector (resp. zero scalar) fields. Note that the user is free to provide $\alpha$ and $\beta$ as constant fields. */ if (alpha.x.i < 0) alpha = unityf; if (lambda.i < 0) { const scalar zeroc[] = 0.; // fixme lambda = zeroc; } /** We need $\alpha$ and $\lambda$ 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. $$\mathbf{u}_f^{n+1} \leftarrow \mathbf{u}_f - \Delta t\alpha\nabla p$$ so that $$\nabla\cdot\mathbf{u}_f^{n+1} = 0$$ This gives the Poisson equation for the pressure $$\nabla\cdot(\alpha\nabla p) = \frac{\nabla\cdot\mathbf{u}_f}{\Delta t}$$ */ trace mgstats project (face vector uf, scalar p, (const) face vector alpha = unityf, double dt = 1., int nrelax = 4) { /** We allocate a local scalar field and compute the divergence of $\mathbf{u}_f$. The divergence is scaled by *dt* so that the pressure has the correct dimension. */ scalar div[]; foreach() { div[] = 0.; foreach_dimension() 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 $$|\nabla\cdot\mathbf{u}_f|\Delta t$$ Given the scaling of the divergence above, this gives */ mgstats mgp = poisson (p, div, alpha, tolerance = TOLERANCE/sq(dt), nrelax = nrelax); /** And compute $\mathbf{u}_f^{n+1}$ using $\mathbf{u}_f$ and $p$. */ foreach_face() uf.x[] -= dt*alpha.x[]*face_gradient_x (p, 0); return mgp; }