Efficient, High Accuracy ADER-WENO Schemes for Hydrodynamics and Divergence-Free Magnetohydrodynamics

TL;DR

This paper presents a new class of high-order finite volume schemes combining ADER and WENO techniques for hyperbolic systems like Euler and MHD equations, demonstrating robustness and accuracy in complex multidimensional shock problems.

Contribution

It introduces a novel formulation of ADER schemes using a local space-time Galerkin approach and develops efficient WENO interpolation methods for structured meshes.

Findings

Schemes achieve their designed accuracy levels.

High-order schemes perform robustly in complex shock scenarios.

Demonstrated effectiveness in 1D, 2D, and 3D tests.

Abstract

The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas. Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We also present a new formulation of the ADER (for Arbitrary Derivative Riemann Problem) schemes that relies on a local continuous space-time Galerkin formulation instead of the usual Cauchy-Kovalewski procedure. The schemes reported here have all been implemented in the RIEMANN framework for computational astrophysics. We demonstrate that the ADER-WENO meet their design accuracies. Several stringent test problems of Euler flows and MHD flows are presented in one, two and three dimensions. Many of our test problems involve near infinite shocks in multiple dimensions and the higher order schemes are shown to perform very robustly and accurately under all conditions.