Energy-conserving finite difference scheme for compressible magnetohydrodynamic flow at low Mach numbers using nonconservative Lorentz force

TL;DR

This paper introduces an energy-conserving finite difference scheme for low Mach number compressible MHD flows that accurately preserves energy and magnetic divergence-free conditions without assuming incompressibility.

Contribution

It develops a novel discretization of the Lorentz force that maintains energy conservation and divergence-free magnetic fields in low Mach number MHD simulations.

Findings

The scheme conserves momentum, magnetic flux, and total energy discretely.

It accurately predicts energy attenuation in magnetic vortex decay.

The method captures effects of compressibility on flow transition to turbulence.

Abstract

In magnetohydrodynamic (MHD) flows, incompressibility is assumed for low Mach numbers. However, even at low Mach numbers, the Mach number influences flow and magnetic fields. Therefore, it is necessary to develop a method that can stably analyze low Mach number compressible MHD flows without using the incompressible assumption. This study constructs an energy-conserving finite difference method to analyze compressible MHD flows at low Mach numbers with the nonconservative Lorentz force. This analysis method discretizes the Lorentz force so that the transformation between conservative and nonconservative forms holds. This scheme simultaneously relaxes velocity, pressure, density, and internal energy, and stable convergence solutions can be obtained. In this study, we analyze four types of models and verify the accuracy and convergence of this numerical method. In the analyses of two- and three-dimensional ideal periodic inviscid MHD flows, it is clarified that momentum, magnetic flux density, and total energy are conserved discretely. The total energy is conserved even in a nonuniform grid. Even without correction for the magnetic flux density, the divergence-free condition of the magnetic flux density is satisfied discretely. Analysis of a Taylor decaying vortex under a magnetic field clarifies that the present numerical method can be applied to incompressible flows and can accurately predict the trend of energy attenuation. In the Orszag-Tang vortex analysis, an increase in Mach number reduces the magnitude of vorticity and current density. In addition, compression work increases more than expansion work, and the influence of compressibility appears. An increase in Mach number slightly delays the transition to turbulent flow. This numerical method has excellent energy conservation properties and can accurately predict energy conversion.