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

Hideki Yanaoka

TL;DR

This work develops an energy-conserving finite-difference framework for compressible MHD at low Mach numbers that employs a nonconservative Lorentz force while preserving the ability to transform between conservative and nonconservative forms. The method uses a spatiotemporal staggered grid and implicit midpoint time integration with simultaneous relaxation to achieve discrete conservation of momentum, magnetic flux, and total energy, as well as discrete satisfaction of Gauss's law for magnetism. By deriving discrete forms for the magnetic energy and magnetic helicity, the approach robustly captures energy exchange between the velocity and magnetic fields across a range of Mach numbers, including the incompressible limit. Verification across inviscid, magnetized vortex, Taylor decaying vortex, and Orszag–Tang vortex problems demonstrates accurate energy budgeting, second-order spatial accuracy, and stability on both uniform and nonuniform grids, with potential for applications in low-Mach MHD control and design.

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.

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

TL;DR

This work develops an energy-conserving finite-difference framework for compressible MHD at low Mach numbers that employs a nonconservative Lorentz force while preserving the ability to transform between conservative and nonconservative forms. The method uses a spatiotemporal staggered grid and implicit midpoint time integration with simultaneous relaxation to achieve discrete conservation of momentum, magnetic flux, and total energy, as well as discrete satisfaction of Gauss's law for magnetism. By deriving discrete forms for the magnetic energy and magnetic helicity, the approach robustly captures energy exchange between the velocity and magnetic fields across a range of Mach numbers, including the incompressible limit. Verification across inviscid, magnetized vortex, Taylor decaying vortex, and Orszag–Tang vortex problems demonstrates accurate energy budgeting, second-order spatial accuracy, and stability on both uniform and nonuniform grids, with potential for applications in low-Mach MHD control and design.

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.
Paper Structure (23 sections, 122 equations, 27 figures)

This paper contains 23 sections, 122 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Modeling and fundamental equations
  3. Fundamental equation
  4. Energy equations
  5. Entropy equation
  6. Magnetic helicity equation
  7. Incompressible limit
  8. Discretization of transport equation
  9. Definitions of finite difference and interpolation operations
  10. Discretization of the Lorentz force
  11. Discretization of mass, momentum, and internal energy equations
  12. Discretization of Faraday's equation
  13. Discretization of the magnetic vector potential equation
  14. Derivation of the magnetic energy equation
  15. Derivation of the total energy equation
  16. ...and 8 more sections

Figures (27)

  • Figure 1: Time variation of total amount of momentum: $\Delta t/(L/U) = 0.01$
  • Figure 2: Time variation of total amount of magnetic flux density: $\Delta t/(L/U) = 0.01$
  • Figure 3: Time variation of momentum error: $\Delta t/(L/U) = 0.01$
  • Figure 4: Time variation of magnetic flux density error: $\Delta t/(L/U) = 0.01$
  • Figure 5: Time variations of total energy and entropy errors: $\Delta t/(L/U) = 0.01$
  • ...and 22 more figures