Implicit-Explict Multistep Methods for Fast-Wave Slow-Wave Problems

Implicit-explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fast-wave, slow-wave problems in which the fast wave is physically insignificant and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapfrog explicit differencing is compared to schemes based on 5 Adams methods or on backward differencing. Two new methods are proposed. The best method appears to be AI22/AB3, which combines a three-time-level A-stable Adams implicit method with the familiar third-order Adams-Bashforth scheme. The implicit part of this scheme, AI22, is not a particularly attractive method as a stand alone implicit scheme, but it appears to have optimal stability properties for this particular IMEX application. The 10 second new method BI22/BX32 appears to have better stability properties for the fast-wave, slow-wave problem than previously explored IMEX backward-differencing combinations, but is nevertheless inferior to AI22/AB3. Both new schemes are fully second order; AI22/AB3 requires only slightly more computation time (about 15%) than trapezoidal-leapfrog methods. 15 The behavior of these schemes is compared theoretically in the context of the simple oscillation equation and also for the linearized equations governing stratified compressible flow. Several schemes are also tested in a fully nonlinear simulation of gravity waves generated by a localized source in a shear flow.