Spectral Deferred Corrections with Fast-wave Slow-wave Splitting

Abstract

The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves (``fast-wave slow-wave problem''). We show that for a scalar test problem with two imaginary eigenvalues $i \lambda_{\text{f}}$, $i \lambda_{\text{s}}$, having $\Delta t ( | \lambda_{\text{f}} | + | \lambda_{\text{s}} | ) < 1$ is sufficient for the fast-wave slow-wave SDC (fwsw-SDC) iteration to converge and that in the limit of infinitely fast waves the convergence rate of the nonsplit version is retained. Stability function and discrete dispersion relation are derived and show that the method is stable for essentially arbitrary fast-wave CFL numbers as long as the slow dynamics are resolved. The method causes little numerical diffusion and its semidiscrete phase speed is accurate also for large wave number modes. Performance is studied for an acoustic-advection problem and for the linearised Boussinesq equations, describing compressible, stratified flow. fwsw-SDC is compared to diagonally implicit Runge--Kutta (DIRK) and implicit-explicit (IMEX) Runge--Kutta methods and found to be competitive in terms of both accuracy and cost.