Spectral Deferred Corrections with Fast-wave Slow-wave Splitting

Abstract

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 iλ f , iλs, having ∆t (|λ f | + |λs|) < 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.Key words. spectral deferred corrections, fast-wave slow-wave splitting, Euler equations, acoustic-advection AMS subject classifications.1. Introduction. For simulations of compressible flow, in particular in numerical weather prediction and climate simulations, the presence of acoustic waves can pose significant numerical challenges to the time integration method. Explicit methods are restricted to inefficiently small steps while fully implicit methods are expensive and can artificially slow down high wave number modes. The fully compressible equations can be replaced by filtered models that do not support sound waves [9,30], but these require solution of a Poisson problem in each step and have difficulties capturing large-scale wave dynamics [8].Therefore, a widely used class of methods are split-explicit integrators: they separate the equation into fast and slow processes which are then integrated with different time step sizes and different (explicit) methods. A popular method of this type is a third-order Runge-Kutta scheme combined with a forward-backward Euler integrator for the acoustic terms [45]. While computationally efficient, split-explicit methods typically require some form of damping for stabilisation [3], which reduces their effective order of accuracy. However, a second-order split-explicit two-step peer method has recently been derived that allows for stable integration of the compressible Euler equations without damping [23].Another form of splitting are semi-implicit methods. They also split the equations into fast and slow parts but then use an implicit method for the fast and an explicit method for the slow part. In many applications, the fast, stiff terms stem from diffusion and/or rapid chemical reactions and methods with IMEX splitting for equations of reaction-diffusion type have been widely studied [24,29,34]. For stratified, compressible flows, however, the fast dynamics are not dif...

Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere

“…IMplicit-EXplicit Spectral Deferred Correction (IMEX SDC) SDC methods have been presented in Dutt et al (2000) and then generalized to methods with different temporal splittings in Minion (2003); Bourlioux et al (2003); Layton and Minion (2004). In the context of fast-wave slow-wave problems, the properties of IMEX SDC schemes have been studied in Ruprecht and Speck (2016). In SDC methods, the interval [t n , t n+1 ] is decomposed into M subintervals using M +1 Gauss-Lobatto temporal nodes, such that t n ≡ t n,0 < t n,1 < · · · < t n,M = t n + ∆t ≡ t n+1 .…”

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

“…While the standard way of solving this is a simplified Newton approach, the more recent development of SDC methods (see the work of Dutt et al) provides an interesting and very flexible alternative. In order to present this approach, we follow the idea of preconditioned Picard iteration, as found for example in other works . The key idea here is to provide a flexible preconditioner based on a simpler quadrature rule for the integrals.…”

“…In order to present this approach, we follow the idea of preconditioned Picard iteration, as found for example in other works. [24][25][26] The key idea here is to provide a flexible preconditioner based on a simpler quadrature rule for the integrals. More precisely, the iteration k is given by…”

Inexact Spectral Deferred Corrections

“…In the last decade, SDC has been applied e.g. to gas dynamics and incompressible or reactive flows [Bouzarth and Minion 2010;Layton and Minion 2004;Minion 2004] as well as to fast-wave slow-wave problems [Ruprecht and Speck 2016], atmospheric modeling [Jia et al 2014], phase-field problems [Feng et al 2015] or particle motions in magnetic fields [Winkel et al 2015]. In addition to its flexibility, SDC has been proven to provide many opportunities for algorithmic and mathematical improvements, e.g.…”

“…• convergence can be accelerated by GMRES [Huang et al 2006] or algebraic preconditioners [Weiser 2014], • high-order implicit-explicit or even multi-implicit splitting is straightforward [Bourlioux et al 2003;Minion 2003;Ruprecht and Speck 2016], • multirate integration allows effective treatment of different time-scales [Bouzarth and Minion 2010;Naumann et al 2018] • inexact spatial solvers enhance time-to-solution [Speck et al 2016a;Weiser and Ghosh 2018].…”