JOURNAL OF COMPUTATIONAL PHYSICS 94, 3&58 (1991) Spectral Method Solution of the Stokes Equations on Nonstaggered Grids MARK R. SCHUMACK AND WILLIAM W. SCHULTZ Depurtment qf Mechanical Engineering and Applied Mechunics, Uniuers~ty qf Michigun, Ann Arbor, Michigan 48109 AND JOHN P. BOYD Department of Atrmwpherir, Oceanic, and Space Science, University of Michigan, Ann Arbor, Michigan 48109 Received June 16, 1989; revised December 28, 1989 The Stokes equations are solved using spectral methods with staggered and nonstaggered grids. Numerous ways to avoid the problem of spurious pressure modes are presented, including new techniques using the pseudospectrdJ method and a method solving the weak form of the governing equations (a variatron on the “spectral element” method developed by Patera). The pseudospectral methods using nonstaggercd grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations. Three test cases arc presented: a formulation with an exact solution, a formulation with homogeneous boundary conditions, and the driven cavity problem. The solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computatmnal solution. :(’ ,991 Academic Press, 1°C 1. INTR000Crr0~ Spectral methods offer high accuracy and are promising tools for fluid dynamics problems requiring high resolution. When solving the equations governing incom- pressible flow in primitive variable form, however, the numerical solution can be polluted by spurious pressures (nontrivial null spaces in the matrix of the discrete problem). Researchers have avoided the spurious pressure problem by solving a Poisson equation for pressure [ 151. A chief concern for the formulations involving a Poisson equation is the pressure boundary conditions [ 12, 15). When the momentum and continuity equations are solved directly, i.e., without a Poisson equation for pressure, staggered grids [2, 4, 9, 10, 13, 161 are often employed to avoid spurious pressures. Staggered grids, however, complicate the programming and can also require the construction of particular solutions for inhomogeneous boundary conditions. 30