IMPLICIT-EXPLICIT METHODS FOR TIME-DEPENDENT PDE'S

URI M. ASCHER?, STEVEN J. RUUTHy , AND BRIAN T.R. WETTONz

Abstract. Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.

For the prototype linear advection-diffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified.

Numerical experiments demonstrate that weak decay of high frequency modes can lead to extra iterations on the finest grid when using multigrid computations with finite difference spatial discretization, and to aliasing when using spectral collocation for spatial discretization. When this behaviour occurs, use of weakly damping schemes such as the popular combination of Crank-Nicolson with second order Adams-Bashforth is discouraged and better alternatives are proposed. Our findings are demonstrated on several examples.

Key words. method of lines, finite differences, spectral methods, aliasing, multigrid, stability region.

AMS subject classifications. 65J15,65M20

1. Introduction. Various methods have been proposed to integrate dynamical systems arising from spatially discretized time-dependent partial differential equations. For problems with terms of different types, implicit-explicit (IMEX) schemes have been often used, especially in conjunction with spectral methods [7, 16]. For convectiondiffusion problems, for example, one would use an explicit scheme for the convection term and an implicit scheme for the diffusion term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.

Consider a time-dependent PDE in which the spatial derivatives have been discretized by central finite differences or by some spectral method. This gives rise to a large system of ODEs in time which typically has the form

_u = f(u) + ?g(u)(1)

where kgk is normalized and ? is a nonnegative parameter. The term f(u) in (1) is some possibly nonlinear term which we do not want to integrate implicitly. This could be because the Jacobian of f(u) is non-symmetric and non-definite and an

? Department of Computer Science, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada. The work of this author was partially supported under NSERC Canada Grant OGP0004306.

y Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada. The work of this author was partially supported by an NSERC Postgraduate Scholarship. z Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada. The work of this author was partially supported under NSERC Canada Grant OGP0122105.