Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - Speed comparisons with Runge-Kutta methods

ADER (Arbitrary DERivative in space and time) methods for the time-evolution of hyperbolic conservation laws have recently generated a fair bit of interest. The ADER time update can be carried out in a single step, which is desirable in many applications. However, prior papers have focused on the theory while downplaying implementation details. The purpose of the present paper is to make ADER schemes accessible by providing two useful formulations of the method as well as their implementation details on three-dimensional structured meshes. We therefore provide a detailed formulation of ADER schemes for conservation laws with non-stiff source terms in nodal as well as modal space along with useful implementation-related details. A good implementation of ADER requires a fast method for transcribing from nodal to modal space and vice versa and we provide innovative transcription strategies that are computationally efficient. We also provide details for the efficient use of ADER schemes in obtaining the numerical flux for conservation laws as well as electric fields for divergence-free magnetohydrodynamics (MHD). An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magnetic fields in MHD is also presented. Several explicit formulae have been provided in all instances for ADER schemes spanning second to fourth orders. The schemes catalogued here have been implemented in the first author's RIEMANN code. The speed of ADER schemes is shown to be almost twice as fast as that of strong stability preserving Runge-Kutta time stepping schemes for all the orders of accuracy that we tested. The modal and nodal ADER schemes have speeds that are within ten percent of each other. When a linearized Riemann solver is used, the third order ADER schemes are half as fast as the second order ADER schemes and the fourth order ADER schemes are a third as fast as the third order ADER schemes. The third order ADER scheme, either with an HLL or linearized Riemann solver, represents an excellent upgrade path for scientists and engineers who are working with a second order Runge-Kutta based total variation diminishing (TVD) scheme. Several stringent test problems have been catalogued. (C) 2012 Elsevier Inc. All rights reserved.