Efficient Implementation of ADER Schemes for Euler and Magnetohydrodynamical Flows on Structured Meshes -- Comparison with Runge-Kutta Methods

TL;DR

This paper presents detailed implementation strategies for ADER schemes applied to Euler and magnetohydrodynamical flows on structured meshes, demonstrating they are nearly twice as fast as traditional Runge-Kutta methods.

Contribution

It provides practical formulations and implementation details of ADER schemes for hyperbolic conservation laws, including MHD, with a focus on efficiency and comparison with Runge-Kutta methods.

Findings

ADER schemes are nearly twice as fast as Runge-Kutta schemes.

Implementation details for ADER schemes on structured meshes are provided.

Efficient strategies for flux and magnetic field calculations in MHD are demonstrated.

Abstract

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 detail. 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. An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magnetic fields in MHD is also presented. 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.