A low-dissipation central scheme for ideal MHD

TL;DR

This paper introduces a low-dissipation central scheme for ideal MHD that improves wave resolution and maintains divergence-free magnetic fields, validated through challenging test cases demonstrating enhanced accuracy and stability.

Contribution

The paper extends a low dissipation central upwind method to ideal MHD, incorporating a constrained transport approach and third-order Runge-Kutta time integration.

Findings

Enhanced resolution of contact discontinuities.

Maintains divergence-free magnetic fields to machine precision.

Achieves experimental second-order accuracy for smooth solutions.

Abstract

Central schemes for conservation laws are Riemann solver free methods which are simple and easy to implement. In recent work for Euler equations [Kurganov & Xin, J. Sci. Comput., 96:56, 2023] their accuracy has been enhanced in terms of better resolution of contact waves. In this paper, we extend this low dissipation central upwind method to the ideal MHD system in one- and two-dimensions. In the two-dimensional case, we separate the variables into two groups: hydrodynamic and magnetic, which are stored at cell centers and faces, respectively. For the the hydrodynamic variables, we apply the low dissipation central upwind scheme while for the magnetic variables, a constrained transport method is used which maintains the divergence-free property of the magnetic field. The time integration is performed with third order strong stability preserving Runge-Kutta scheme. To validate the proposed scheme, we apply this method to several challenging test cases. The results show that the LDCU correction term plays a useful role at the contact discontinuity and enhances the resolution of waves. We also observe experimental second-order accuracy for smooth solutions and the divergence-free condition is maintained to machine precision.