Implicit-Explicit Methods for Time-Dependent Partial Differential Equations

TLDR

Implicit–explicit (IMEX) schemes, commonly pairing implicit diffusion with explicit convection, are widely used for time integration of diffusion–convection and reaction–diffusion PDEs, often with spectral methods. This study systematically evaluates IMEX performance, introduces improved schemes, and examines their effectiveness in fast multigrid algorithms and aliasing reduction for spectral methods. A stability analysis of first‑ through fourth‑order multistep IMEX schemes is carried out on a prototype linear advection–diffusion equation. The analysis identifies stable, large‑time‑step schemes that properly damp high‑frequency errors; numerical tests show weak damping causes extra multigrid iterations or aliasing, leading to the recommendation of stronger damping alternatives over the popular Crank–Nicolson/Adams–Bashforth pair, with results confirmed on multiple examples.

Abstract

Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (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 behavior 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.