Stokes integrators

stokes_integrators

This module provides a fine-tuned solver class for the Stokes system of conservation equations and a function to automatically set the associated null spaces. Users instantiate the StokesSolver class by providing relevant parameters and call the solve method to request a solver update.

iterative_stokes_solver_parameters = {'mat_type': 'matfree', 'ksp_type': 'preonly', 'pc_type': 'fieldsplit', 'pc_fieldsplit_type': 'schur', 'pc_fieldsplit_schur_type': 'full', 'fieldsplit_0': {'ksp_type': 'cg', 'ksp_rtol': 1e-05, 'ksp_max_it': 1000, 'pc_type': 'python', 'pc_python_type': 'gadopt.SPDAssembledPC', 'assembled_pc_type': 'gamg', 'assembled_mg_levels_pc_type': 'sor', 'assembled_pc_gamg_threshold': 0.01, 'assembled_pc_gamg_square_graph': 100, 'assembled_pc_gamg_coarse_eq_limit': 1000, 'assembled_pc_gamg_mis_k_minimum_degree_ordering': True}, 'fieldsplit_1': {'ksp_type': 'fgmres', 'ksp_rtol': 0.0001, 'ksp_max_it': 200, 'pc_type': 'python', 'pc_python_type': 'firedrake.MassInvPC', 'Mp_pc_type': 'ksp', 'Mp_ksp_ksp_rtol': 1e-05, 'Mp_ksp_ksp_type': 'cg', 'Mp_ksp_pc_type': 'sor'}} module-attribute

Default iterative solver parameters for solution of stokes system.

We configure the Schur complement approach as described in Section of 4.3 of Davies et al. (2022), using PETSc's fieldsplit preconditioner type, which provides a class of preconditioners for mixed problems that allows a user to apply different preconditioners to different blocks of the system.

The fieldsplit_0 entries configure solver options for the velocity block. The linear systems associated with this matrix are solved using a combination of the Conjugate Gradient (cg) method and an algebraic multigrid preconditioner (GAMG).

The fieldsplit_1 entries contain solver options for the Schur complement solve itself. For preconditioning, we approximate the Schur complement matrix with a mass matrix scaled by viscosity, with the viscosity provided through the optional mu keyword argument to Stokes solver. Since this preconditioner step involves an iterative solve, the Krylov method used for the Schur complement needs to be of flexible type, and we use FGMRES by default.

We note that our default solver parameters can be augmented or adjusted by accessing the solver_parameter dictionary, for example:

   stokes_solver.solver_parameters['fieldsplit_0']['ksp_converged_reason'] = None
   stokes_solver.solver_parameters['fieldsplit_0']['ksp_rtol'] = 1e-3
   stokes_solver.solver_parameters['fieldsplit_0']['assembled_pc_gamg_threshold'] = -1
   stokes_solver.solver_parameters['fieldsplit_1']['ksp_converged_reason'] = None
   stokes_solver.solver_parameters['fieldsplit_1']['ksp_rtol'] = 1e-2

Note

G-ADOPT defaults to iterative solvers in 3-D.

direct_stokes_solver_parameters = {'mat_type': 'aij', 'ksp_type': 'preonly', 'pc_type': 'lu', 'pc_factor_mat_solver_type': 'mumps'} module-attribute

Default direct solver parameters for solution of Stokes system.

Configured to use LU factorisation, using the MUMPS library.

Note

G-ADOPT defaults to direct solvers in 2-D.

newton_stokes_solver_parameters = {'snes_type': 'newtonls', 'snes_linesearch_type': 'l2', 'snes_max_it': 100, 'snes_atol': 1e-10, 'snes_rtol': 1e-05} module-attribute

Default solver parameters for non-linear systems.

We use a setup based on Newton's method (newtonls) with a secant line search over the L2-norm of the function.

StokesSolver(z, T, approximation, bcs={}, mu=1, quad_degree=6, solver_parameters=None, J=None, constant_jacobian=False, **kwargs)

Solves the Stokes system in place.

Parameters:

Name	Type	Description	Default
z	Function	Firedrake function representing mixed Stokes system	required
T	Function	Firedrake function representing temperature	required
approximation	BaseApproximation	Approximation describing system of equations	required
bcs	dict[int, dict[str, Number]]	Dictionary of identifier-value pairs specifying boundary conditions	{}
mu	Function | Number	Firedrake function representing dynamic viscosity	1
quad_degree	int	Quadrature degree. Default value is 2p + 1, where p is the polynomial degree of the trial space	6
solver_parameters	Optional[dict[str, str | Number] | str]	Either a dictionary of PETSc solver parameters or a string specifying a default set of parameters defined in G-ADOPT	None
J	Optional[Function]	Firedrake function representing the Jacobian of the system	None
constant_jacobian	bool	Whether the Jacobian of the system is constant	False

Source code in g-adopt/gadopt/stokes_integrators.py

def __init__(
    self,
    z: fd.Function,
    T: fd.Function,
    approximation: BaseApproximation,
    bcs: dict[int, dict[str, Number]] = {},
    mu: fd.Function | Number = 1,
    quad_degree: int = 6,
    solver_parameters: Optional[dict[str, str | Number] | str] = None,
    J: Optional[fd.Function] = None,
    constant_jacobian: bool = False,
    **kwargs,
):
    self.Z = z.function_space()
    self.mesh = self.Z.mesh()
    self.test = fd.TestFunctions(self.Z)
    self.equations = StokesEquations(self.Z, self.Z, quad_degree=quad_degree,
                                     compressible=approximation.compressible)
    self.solution = z
    self.solution_old = None
    self.approximation = approximation
    self.mu = ensure_constant(mu)
    self.J = J
    self.constant_jacobian = constant_jacobian
    self.linear = not depends_on(self.mu, self.solution)

    self.solver_kwargs = kwargs
    u, p = fd.split(self.solution)
    self.k = upward_normal(self.Z.mesh())
    self.fields = {
        'velocity': u,
        'pressure': p,
        'viscosity': self.mu,
        'interior_penalty': fd.Constant(2.0),  # allows for some wiggle room in imposition of weak BCs
                                               # 6.25 matches C_ip=100. in "old" code for Q2Q1 in 2d.
        'source': self.approximation.buoyancy(p, T) * self.k,
        'rho_continuity': self.approximation.rho_continuity(),
    }

    self.weak_bcs = {}
    self.strong_bcs = []
    for id, bc in bcs.items():
        weak_bc = {}
        for bc_type, value in bc.items():
            if bc_type == 'u':
                self.strong_bcs.append(fd.DirichletBC(self.Z.sub(0), value, id))
            elif bc_type == 'ux':
                self.strong_bcs.append(fd.DirichletBC(self.Z.sub(0).sub(0), value, id))
            elif bc_type == 'uy':
                self.strong_bcs.append(fd.DirichletBC(self.Z.sub(0).sub(1), value, id))
            elif bc_type == 'uz':
                self.strong_bcs.append(fd.DirichletBC(self.Z.sub(0).sub(2), value, id))
            else:
                weak_bc[bc_type] = value
        self.weak_bcs[id] = weak_bc

    self.F = 0
    for test, eq, u in zip(self.test, self.equations, fd.split(self.solution)):
        self.F -= eq.residual(test, u, u, self.fields, bcs=self.weak_bcs)

    if isinstance(solver_parameters, dict):
        self.solver_parameters = solver_parameters
    else:
        if self.linear:
            self.solver_parameters = {"snes_type": "ksponly"}
        else:
            self.solver_parameters = newton_stokes_solver_parameters.copy()

        if INFO >= log_level:
            self.solver_parameters["snes_monitor"] = None

        if isinstance(solver_parameters, str):
            match solver_parameters:
                case "direct":
                    self.solver_parameters.update(direct_stokes_solver_parameters)
                case "iterative":
                    self.solver_parameters.update(
                        iterative_stokes_solver_parameters
                    )
                case _:
                    raise ValueError(
                        f"Solver type '{solver_parameters}' not implemented."
                    )
        elif self.mesh.topological_dimension() == 2 and self.mesh.cartesian:
            self.solver_parameters.update(direct_stokes_solver_parameters)
        else:
            self.solver_parameters.update(iterative_stokes_solver_parameters)

            if DEBUG >= log_level:
                self.solver_parameters['fieldsplit_0']['ksp_converged_reason'] = None
                self.solver_parameters['fieldsplit_1']['ksp_monitor'] = None
            elif INFO >= log_level:
                self.solver_parameters['fieldsplit_1']['ksp_converged_reason'] = None

    # solver object is set up later to permit editing default solver parameters specified above
    self._solver_setup = False

setup_solver()

Sets up the solver.

Source code in g-adopt/gadopt/stokes_integrators.py

def setup_solver(self):
    """Sets up the solver."""
    # mu used in MassInvPC:
    mu_over_rho = self.mu / self.approximation.rho_continuity()
    if self.constant_jacobian:
        z_tri = fd.TrialFunction(self.Z)
        F_stokes_lin = fd.replace(self.F, {self.solution: z_tri})
        a, L = fd.lhs(F_stokes_lin), fd.rhs(F_stokes_lin)
        self.problem = fd.LinearVariationalProblem(a, L, self.solution,
                                                   bcs=self.strong_bcs,
                                                   constant_jacobian=True)
        self.solver = fd.LinearVariationalSolver(self.problem,
                                                 solver_parameters=self.solver_parameters,
                                                 options_prefix=self.name,
                                                 appctx={"mu": mu_over_rho},
                                                 **self.solver_kwargs)
    else:
        self.problem = fd.NonlinearVariationalProblem(self.F, self.solution,
                                                      bcs=self.strong_bcs, J=self.J)
        self.solver = fd.NonlinearVariationalSolver(self.problem,
                                                    solver_parameters=self.solver_parameters,
                                                    options_prefix=self.name,
                                                    appctx={"mu": mu_over_rho},
                                                    **self.solver_kwargs)
    self._solver_setup = True

solve()

Solves the system.

Source code in g-adopt/gadopt/stokes_integrators.py

def solve(self):
    """Solves the system."""
    if not self._solver_setup:
        self.setup_solver()
    self.solution_old = self.solution.copy(deepcopy=True)
    self.solver.solve()

create_stokes_nullspace(Z, closed=True, rotational=False, translations=None, ala_approximation=None, top_subdomain_id=None)

Create a null space for the mixed Stokes system.

Parameters:

Name	Type	Description	Default
Z	WithGeometry	Firedrake mixed function space associated with the Stokes system	required
closed	bool	Whether to include a constant pressure null space	True
rotational	bool	Whether to include all rotational modes	False
translations	Optional[list[int]]	List of translations to include	None
ala_approximation	Optional[AnelasticLiquidApproximation]	AnelasticLiquidApproximation for calculating (non-constant) right nullspace	None
top_subdomain_id	Optional[str | int]	Boundary id of top surface. Required when providing ala_approximation.	None

Returns:

Type	Description
MixedVectorSpaceBasis	A Firedrake mixed vector space basis incorporating the null space components

Source code in g-adopt/gadopt/stokes_integrators.py

def create_stokes_nullspace(
    Z: fd.functionspaceimpl.WithGeometry,
    closed: bool = True,
    rotational: bool = False,
    translations: Optional[list[int]] = None,
    ala_approximation: Optional[AnelasticLiquidApproximation] = None,
    top_subdomain_id: Optional[str | int] = None,
) -> fd.nullspace.MixedVectorSpaceBasis:
    """Create a null space for the mixed Stokes system.

    Arguments:
      Z: Firedrake mixed function space associated with the Stokes system
      closed: Whether to include a constant pressure null space
      rotational: Whether to include all rotational modes
      translations: List of translations to include
      ala_approximation: AnelasticLiquidApproximation for calculating (non-constant) right nullspace
      top_subdomain_id: Boundary id of top surface. Required when providing
                        ala_approximation.

    Returns:
      A Firedrake mixed vector space basis incorporating the null space components

    """
    # ala_approximation and top_subdomain_id are both needed when calculating right nullspace for ala
    if (ala_approximation is None) != (top_subdomain_id is None):
        raise ValueError("Both ala_approximation and top_subdomain_id must be provided, or both must be None.")

    X = fd.SpatialCoordinate(Z.mesh())
    dim = len(X)
    V, W = Z.subfunctions

    if rotational:
        if dim == 2:
            rotV = fd.Function(V).interpolate(fd.as_vector((-X[1], X[0])))
            basis = [rotV]
        elif dim == 3:
            x_rotV = fd.Function(V).interpolate(fd.as_vector((0, -X[2], X[1])))
            y_rotV = fd.Function(V).interpolate(fd.as_vector((X[2], 0, -X[0])))
            z_rotV = fd.Function(V).interpolate(fd.as_vector((-X[1], X[0], 0)))
            basis = [x_rotV, y_rotV, z_rotV]
        else:
            raise ValueError("Unknown dimension")
    else:
        basis = []

    if translations:
        for tdim in translations:
            vec = [0] * dim
            vec[tdim] = 1
            basis.append(fd.Function(V).interpolate(fd.as_vector(vec)))

    if basis:
        V_nullspace = fd.VectorSpaceBasis(basis, comm=Z.mesh().comm)
        V_nullspace.orthonormalize()
    else:
        V_nullspace = V

    if closed:
        if ala_approximation:
            p = ala_right_nullspace(W=W, approximation=ala_approximation, top_subdomain_id=top_subdomain_id)
            p_nullspace = fd.VectorSpaceBasis([p], comm=Z.mesh().comm)
            p_nullspace.orthonormalize()
        else:
            p_nullspace = fd.VectorSpaceBasis(constant=True, comm=Z.mesh().comm)
    else:
        p_nullspace = W

    return fd.MixedVectorSpaceBasis(Z, [V_nullspace, p_nullspace])

ala_right_nullspace(W, approximation, top_subdomain_id)

Compute pressure nullspace for Anelastic Liquid Approximation.

Parameters:

Name	Type	Description	Default
W	WithGeometry	pressure function space	required
approximation	AnelasticLiquidApproximation	AnelasticLiquidApproximation with equation parameters	required
top_subdomain_id	str | int	boundary id of top surface	required

Returns:

Type	Description
	pressure nullspace solution

To obtain the pressure nullspace solution for the Stokes equation in Anelastic Liquid Approximation, which includes a pressure-dependent buoyancy term, we try to solve the equation:

\[ -nabla p + g "Di" rho chi c_p/(c_v gamma) hatk p = 0 \]

Taking the divergence:

\[ -nabla * nabla p + nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) = 0, \]

then testing it with q:

\[ int_Omega -q nabla * nabla p dx + int_Omega q nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) dx = 0 \]

followed by integration by parts:

\[ int_Gamma -bb n * q nabla p ds + int_Omega nabla q cdot nabla p dx + int_Gamma bb n * hatk q g "Di" rho chi c_p/(c_v gamma) p dx - int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0 \]

This elliptic equation can be solved with natural boundary conditions by imposing our original equation above, which eliminates all boundary terms:

\[ int_Omega nabla q * nabla p dx - int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0. \]

However, if we do so on all boundaries we end up with a system that has the same nullspace, as the one we are after (note that we ended up merely testing the original equation with \(nabla q\)). Instead we use the fact that the gradient of the null mode is always vertical, and thus the null mode is constant at any horizontal level (geoid), specifically the top surface. Choosing any nonzero constant for this surface fixes the arbitrary scalar multiplier of the null mode. We choose the value of one and apply it as a Dirichlet boundary condition.

Note that this procedure does not necessarily compute the exact nullspace of the discretised Stokes system. In particular, since not every test function \(v in V\), the velocity test space, can be written as \(v=nabla q\) with \(q in W\), the pressure test space, the two terms do not necessarily exactly cancel when tested with \(v\) instead of \(nabla q\) as in our final equation. However, in practice the discrete error appears to be small enough, and providing this nullspace gives an improved convergence of the iterative Stokes solver.

Source code in g-adopt/gadopt/stokes_integrators.py

def ala_right_nullspace(
        W: fd.functionspaceimpl.WithGeometry,
        approximation: AnelasticLiquidApproximation,
        top_subdomain_id: str | int):
    r"""Compute pressure nullspace for Anelastic Liquid Approximation.

        Arguments:
          W: pressure function space
          approximation: AnelasticLiquidApproximation with equation parameters
          top_subdomain_id: boundary id of top surface

        Returns:
          pressure nullspace solution

        To obtain the pressure nullspace solution for the Stokes equation in Anelastic Liquid Approximation,
        which includes a pressure-dependent buoyancy term, we try to solve the equation:

        $$
          -nabla p + g "Di" rho chi c_p/(c_v gamma) hatk p = 0
        $$

        Taking the divergence:

        $$
          -nabla * nabla p + nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) = 0,
        $$

        then testing it with q:

        $$
            int_Omega -q nabla * nabla p dx + int_Omega q nabla * (g "Di" rho chi c_p/(c_v gamma) hatk p) dx = 0
        $$

        followed by integration by parts:

        $$
            int_Gamma -bb n * q nabla p ds + int_Omega nabla q cdot nabla p dx +
            int_Gamma bb n * hatk q g "Di" rho chi c_p/(c_v gamma) p dx -
            int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0
        $$

        This elliptic equation can be solved with natural boundary conditions by imposing our original equation above, which eliminates
        all boundary terms:

        $$
          int_Omega nabla q * nabla p dx - int_Omega nabla q * hatk g "Di" rho chi c_p/(c_v gamma) p dx = 0.
        $$

        However, if we do so on all boundaries we end up with a system that has the same nullspace, as the one we are after (note that
        we ended up merely testing the original equation with $nabla q$). Instead we use the fact that the gradient of the null mode
        is always vertical, and thus the null mode is constant at any horizontal level (geoid), specifically the top surface. Choosing
        any nonzero constant for this surface fixes the arbitrary scalar multiplier of the null mode. We choose the value of one
        and apply it as a Dirichlet boundary condition.

        Note that this procedure does not necessarily compute the exact nullspace of the *discretised* Stokes system. In particular,
        since not every test function $v in V$, the velocity test space, can be written as $v=nabla q$ with $q in W$, the
        pressure test space, the two terms do not necessarily exactly cancel when tested with $v$ instead of $nabla q$ as in our
        final equation. However, in practice the discrete error appears to be small enough, and providing this nullspace gives
        an improved convergence of the iterative Stokes solver.
    """
    W = fd.FunctionSpace(mesh=W.mesh(), family=W.ufl_element())
    q = fd.TestFunction(W)
    p = fd.Function(W, name="pressure_nullspace")

    # Fix the solution at the top boundary
    bc = fd.DirichletBC(W, 1., top_subdomain_id)

    F = fd.inner(fd.grad(q), fd.grad(p)) * fd.dx

    k = upward_normal(W.mesh())

    F += - fd.inner(fd.grad(q), k * approximation.dbuoyancydp(p, fd.Constant(1.0)) * p) * fd.dx

    fd.solve(F == 0, p, bcs=bc)
    return p