A high-order finite difference WENO scheme for ideal magnetohydrodynamics on curvilinear meshes

TL;DR

This paper introduces a high-order finite difference WENO scheme for ideal magnetohydrodynamics on curvilinear meshes, combining Taylor expansion-based flux computation with Riemann solvers, and demonstrates its effectiveness on benchmark problems.

Contribution

It develops a novel high-order WENO scheme using an alternative flux formulation for MHD on curvilinear meshes, extending Riemann solvers like HLLD to complex geometries.

Findings

The scheme achieves high accuracy on benchmark problems.

Low dissipative Riemann solvers improve solution quality.

The method effectively handles complex boundary conditions.

Abstract

A high-order finite difference numerical scheme is developed for the ideal magnetohydrodynamic equations based on an alternative flux formulation of the weighted essentially non-oscillatory (WENO) scheme. It computes a high-order numerical flux by a Taylor expansion in space, with the lowest-order term solved from a Riemann solver and the higher-order terms constructed from physical fluxes by limited central differences. The scheme coupled with several Riemann solvers, including a Lax-Friedrichs solver and HLL-type solvers, is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems. In particular, a HLLD solver on Cartesian meshes is extended to curvilinear meshes with proper modifications. A numerical boundary condition for the perfect electrical conductor (PEC) boundary is derived for general geometry and verified through a bow shock flow. Numerical results also confirm the advantages of using low dissipative Riemann solvers in the current framework.