Efficient Variable-Coefficient Finite-Volume Stokes Solvers

Abstract

Abstract. We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity-pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595 ], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

Fluctuating hydrodynamics of reactive liquid mixtures

“…We note that our FHD model of multispecies diffusion is constructed so that it reproduces the correct Einstein distribution under Stirling's approximation, i.e., our model is consistent with (27) and (28). Hence, the combined chemistry and FHD model is expected to give the correct equilibrium distribution, as long as there are sufficiently many molecules of all species in each cell to justify the continuous approximation.…”

“…of viscous dissipation to the divergence-free constraint on velocity, to simultaneously solve for the velocity and mechanical pressure. To solve the Stokes system we use a variable-coefficient (tensor) multigrid-preconditioned GMRES (generalized minimal residual) solver [28], as we have done previously [3,26]. The difference between the predictor and corrector Stokes solves is the temporal discretization of the advective term (explicit vs. trapezoidal) and the forcing term (explicit vs. midpoint); both Stokes solves are required for second-order deterministic accuracy.…”

Preconditioners for Two-Phase Incompressible Navier--Stokes Flow

“…Before addressing the appropriate choice in the Stokes case, we must be clear on the form of the viscous term used. In the literature different forms are considered, including −∇ · (µ ∇ u) in [43], −∇ · (µ Du) in [22,41], and −∇ · (2µ Du) in [23,10]. The overall scaling of the latter term contains an additional factor of two compared with the former choices, as such an appropriate Schur complement approximation for the Stokes problem differs by a factor of a half.…”

“…The overall scaling of the latter term contains an additional factor of two compared with the former choices, as such an appropriate Schur complement approximation for the Stokes problem differs by a factor of a half. The distinction between different forms of the viscous term used (including incorporating bulk viscosity) and the effect on the Schur complement is discussed further in [10]. Since we use the latter choice above, we present results for this case, including a factor of two where appropriate as compared with the cited references.…”

“…In the two-phase case, the scaling with density and viscosity must be incorporated within the integral definition of the matrices and so we obtain (31). An inverse Schur complement approximation very similar to (31), appropriate for finite volume Stokes solvers, is given in [10]. Here the MAC discretisation on staggered grids is employed.…”