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 hyper-bolic conservation laws have recently generated a fair bit of interest. The ADER time updatecan be carried out in a single step, which is desirable in many applications. However, priorpapers have focused on the theory while downplaying implementation details. The pur-pose of the present paper is to make ADER schemes accessible by providing two useful for-mulations of the method as well as their implementation details on three-dimensionalstructured meshes. We therefore provide a detailed formulation of ADER schemes for con-servation laws with non-stiff source terms in nodal as well as modal space along with use-ful implementation-related details. A good implementation of ADER requires a fast methodfor transcribing from nodal to modal space and vice versa and we provide innovative tran-scription strategies that are computationally efficient. We also provide details for the effi-cient use of ADER schemes in obtaining the numerical flux for conservation laws as well aselectric fields for divergence-free magnetohydrodynamics (MHD). An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magneticfields in MHD is also presented. Several explicit formulae have been provided in allinstances for ADER schemes spanning second to fourth orders.The schemes catalogued here have been implemented in the first author’s RIEMANNcode. The speed of ADER schemes is shown to be almost twice as fast as that of strong sta-bility preserving Runge–Kutta time stepping schemes for all the orders of accuracy that wetested. The modal and nodal ADER schemes have speeds that are within ten percent of eachother. When a linearized Riemann solver is used, the third order ADER schemes are half asfast as the second order ADER schemes and the fourth order ADER schemes are a third asfast as the third order ADER schemes. The third order ADER scheme, either with an HLL orlinearized Riemann solver, represents an excellent upgrade path for scientists and engi-neers who are working with a second order Runge–Kutta based total variation diminishing(TVD) scheme. Several stringent test problems have been catalogued. © 2012 Elsevier Inc. All rights reserved.