Parallel-in-time high-order multiderivative IMEX solvers

Abstract

In this work, we present a novel class of parallelizable high-order time integration schemes for the approximate solution of additive ODEs. The methods achieve high order through a combination of a suitable quadrature formula involving multiple derivatives of the ODE's right-hand side and a predictor-corrector ansatz. The latter approach is designed in such a way that parallelism in time is made possible. We present thorough analysis as well as numerical results that showcase scaling opportunities of methods from this class of solvers.

Two-derivative deferred correction time discretization for the discontinuous Galerkin method

“…In this section, the semi-discrete in time formulation of the novel scheme is reviewed. For that purpose, we briefly recall the serial algorithm from [20] and [21] and slightly modify it. The algorithm describes a predictor-corrector approach with k max correction steps to approximate a two-derivative Hermite-Birkhoff Runge-Kutta method of order q and is therefore labeled as HBPC(q, k max ).…”

“…The algorithm describes a predictor-corrector approach with k max correction steps to approximate a two-derivative Hermite-Birkhoff Runge-Kutta method of order q and is therefore labeled as HBPC(q, k max ). As we do not use an IMEX splitting in this paper as it is done in [20,21], the predictor is modified such that a fourth-order two-point Hermite-Birkhoff Runge-Kutta method is successively used to obtain the predicted solution. Please note that for convenience, we stick with the name HBPC(q, k max ) although we use a different predictor than in the original publication [21].…”

“…As we do not use an IMEX splitting in this paper as it is done in [20,21], the predictor is modified such that a fourth-order two-point Hermite-Birkhoff Runge-Kutta method is successively used to obtain the predicted solution. Please note that for convenience, we stick with the name HBPC(q, k max ) although we use a different predictor than in the original publication [21].…”

“…(A.1)-(A.3). More details can be found in [21]. These Butcher tableaux define a quadrature formula I l of order q through…”

“…In [20], a fourth order accurate two-derivative deferred correction time discretization method has been presented. Recently, it has been combined with the idea of using multiple stages to obtain higher orders of accuracy [21]. There it has been shown that this class of schemes can be used to solve stiff ODEs while obtaining -at least in principle -arbitrary orders of accuracy.…”