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A low-dissipation central scheme for ideal MHD

Yu-Chen Cheng, Praveen Chandrashekar, Christian Klingenberg

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.

A low-dissipation central scheme for ideal MHD

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.
Paper Structure (22 sections, 68 equations, 16 figures, 3 tables, 1 algorithm)

This paper contains 22 sections, 68 equations, 16 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. 1-D LDCU MHD scheme
  3. 1-D LDCU scheme for the MHD system
  4. 1-D LDCU MHD semi-discrete scheme
  5. 2-D LDCU MHD scheme
  6. 2-D LDCU scheme
  7. Upwind constrained transport HLL method
  8. Numerical results
  9. 1-D experiments
  10. Brio-Wu shock tube problem
  11. Dai & Woodward shock tube problem
  12. Ryu-Jones problem
  13. 2-D experiments
  14. Balsara vortex test
  15. Smooth sine wave
  16. ...and 7 more sections

Figures (16)

  • Figure 2.1: Three steps of the LDCU method, following LDCU2.
  • Figure 2.2: Smooth (green) and unsmooth (red) region. This figure is inspired by Figure 3 in LDCU1.
  • Figure 4.1: Example \ref{['sec.1DBW']} Brio-Wu shock tube problem : Density with 800 grid points, in comparison with the scheme without LDCU projection step.
  • Figure 4.2: Example \ref{['sec.1DDW']} Dai & Woodward shock tube problem: Density with 512 grid points. Compared with the scheme without LDCU correction.
  • Figure 4.3: Example \ref{['sec.1DRJ']} Ryu-Jones problem: Density with 516 grid points. Compared with the scheme without LDCU correction.
  • ...and 11 more figures

Theorems & Definitions (2)

  • Remark 2.1
  • Remark 3.1