Higher-order magnetohydrodynamic numerics

TL;DR

This paper discusses advanced finite-volume numerical methods for magnetohydrodynamics, emphasizing higher-order schemes, their challenges, and improvements, demonstrated through simulations of shocked systems and turbulence.

Contribution

It introduces a fourth-order MHD solver and explores issues, algorithms, and refinements specific to higher-order numerics in fluid simulations.

Findings

Successful implementation of a fourth-order MHD solver

Enhanced accuracy in simulating strongly shocked systems

Insights into turbulence decay in compressible flows

Abstract

In this chapter, we aim at presenting the basic techniques necessary to go beyond the widely accepted paradigm of second-order numerics. We specifically focus on finite-volume schemes for hyperbolic conservation laws occuring in fluid approximations such as the equations of ideal magnetohydrodynamics or the Euler equations of gas dynamics. For the sake of clarity, a simple fourth-order ideal magnetohydrodynamic (MHD) solver which allows to simulate strongly shocked systems serves as an instructive example. Issues that only or mainly arise in the world of higher-order numerics are given specific focus. Alternative algorithms as well as refinements and improvements are dicussed and are referenced to in the literature. As an example of application, some results on decaying compressible turbulence are presented.