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Efficient Variable-Coefficient Finite-Volume Stokes Solvers

Published online by Cambridge University Press:  03 June 2015

Mingchao Cai*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Andy Nonaka*
Affiliation:
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
John B. Bell*
Affiliation:
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Boyce E. Griffith*
Affiliation:
Leon H. Charney Division of Cardiology, Department of Medicine, New York University School of Medicine, NY, USA
Aleksandar Donev*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
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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; 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.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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