Spectral deferred corrections with fast-wave slow-wave splitting

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 iλ_fast, iλ_slow, having Δt(|λ_fast|+|λ_slow|)<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 non-split 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 semi-discrete 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 a diagonally implicit Runge-Kutta (DIRK) and IMEX Runge-Kutta (IMEX) method and found to be competitive in terms of both accuracy and cost.