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Implicit-explicit methods for time-dependent PDE’s Ruuth, Steven J.

Abstract

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 used, especially in conjunction with spectral methods. For convection-diffusion problems, for example, one would use an explicit scheme for the convection term and an implicit scheme for thediffusion 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 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.

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