A Divergence‐free Upwind Code for Multidimensional Magnetohydrodynamic Flows

Abstract

A description is given for preserving ∇ · B = 0 in a magnetohydrodynamic (MHD) code that employs the upwind, Total Variation Diminishing (TVD) scheme and the Strang-type operator splitting for multi-dimensionality. The method is based on the staggered mesh technique to constrain the transport of magnetic field: the magnetic field components are defined at grid interfaces with their advective fluxes on grid edges, while other quantities are defined at grid centers. The magnetic field at grid centers for the upwind step is calculated by interpolating the values from grid interfaces. The advective fluxes on grid edges for the magnetic field evolution are calculated from the upwind fluxes at grid interfaces. Then, the magnetic field can be maintained with ∇ · B = 0 exactly, if this is so initially, while the upwind scheme is used for the update of fluid quantities. The correctness of the code is demonstrated through tests comparing numerical solutions either with analytic solutions or with numerical solutions from the code using an explicit divergence-cleaning method. Also the robustness is shown through tests involving realistic astrophysical problems.

The stellar atmosphere simulation codeBifrost

“…Figure 3 is created to be directly compared with Fig. 3 of Ryu et al (1998) who use a upwind, total variation diminishing code, and even though there are small differences it is hard to tell which code is superior in spite of the two codes being fundamentally different.…”

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

“…Toth [45] found the flux constrained transport method was one of the most accurate schemes that he tested. This approach has been combined with various shock-capturing schemes by different researchers [6,46,47,41,[48][49][50][51][52]. In this paper, the constrained transport method proposed by Balsara and Spicer [41] is adopted.…”

Stable upwind schemes for the magnetic induction equation

“…In this approach, the velocity and magnetic fields in the x-direction are centered on the cell edges in the x-direction and the velocity and magnetic fields in the y-direction are centered on the cell-edges in the y-direction. This approach has been proposed in number a papers including [1,6,7,16,17] and details can be found in these references.…”