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A structure-preserving semi-implicit IMEX finite volume scheme for ideal magnetohydrodynamics at all Mach and Alfvén numbers

Walter Boscheri, Andrea Thomann

Abstract

We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit-EXplicit Runge-Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfvén Mach number regimes where the performance and the stability of the new scheme is assessed.

A structure-preserving semi-implicit IMEX finite volume scheme for ideal magnetohydrodynamics at all Mach and Alfvén numbers

Abstract

We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit-EXplicit Runge-Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfvén Mach number regimes where the performance and the stability of the new scheme is assessed.
Paper Structure (36 sections, 62 equations, 13 figures, 3 tables)

This paper contains 36 sections, 62 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Scaling of the ideal MHD equations
  4. Novel 3-split form of the ideal MHD equations
  5. Numerical scheme
  6. Space and time computational domain.
  7. Time semi-discrete scheme in one space dimension
  8. Time semi-discrete scheme in multiple space dimensions
  9. Second order time discretization
  10. Discrete spatial operators
  11. Numerical flux operator $\vmathbb{F}(f(q))$.
  12. Central flux operator $\vmathbb{B}(f(q))$.
  13. Differential operator $\vmathbb{H}(h,q)$.
  14. Gradient operator $\vmathbb{G}(q)$.
  15. Curl operator $\vmathbb{C}(\mathbf{q})$.
  16. ...and 21 more sections

Figures (13)

  • Figure 1: MHD traveling vortex. Time evolution of the ratio between the material time step of the SIFV-EB scheme and the magneto-sonic time step of a fully explicit finite volume scheme for different values of the background density $\rho_0$.
  • Figure 2: Riemann problems. RP1 at time $t_f=0.1$ (top), RP2 at time $t_f=0.2$ (middle) and RP3 at time $t_f=0.15$ (bottom). Left: density $\rho$. Center: magnetic field component $B_y$. Right: pressure $p$.
  • Figure 3: Riemann problems. RP4 at time $t_f=0.16$, RP5 at time $t_f=0.04$, RP6 at time $t_f=0.03$ and RP7 at time $t_f=0.2$ (from top to bottom). Left: density $\rho$. Center: magnetic field component $B_y$. Right: pressure $p$.
  • Figure 4: Riemann problems. Time evolution of the ratio between the material time step of the SIFV-EB scheme and the magneto-sonic time step of a fully explicit finite volume scheme for all the Riemann problems RP1--RP7.
  • Figure 5: Blast problem at time $t_f=0.01$. Results for the logarithm (base 10) of density and pressure, magnitude of the velocity and magnetic pressure (from top left to bottom right).
  • ...and 8 more figures