A Solution Accurate, Efficient and Stable Unsplit Staggered Mesh Scheme for Three Dimensional Magnetohydrodynamics

TL;DR

This paper extends a 2D unsplit staggered mesh scheme to 3D for magnetohydrodynamics, introducing two algorithms that improve stability, efficiency, and accuracy in simulating magnetic fields.

Contribution

The paper develops a full 3D unsplit staggered mesh scheme with novel algorithms that enhance stability and efficiency for MHD simulations, including a new constrained transport method.

Findings

The full 3D CTU scheme achieves stability with CFL numbers close to unity.

The algorithms reduce computational cost by minimizing Riemann solves.

The methods effectively maintain in-plane magnetic field dynamics.

Abstract

In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D magnetohydrodynamics (MHD) (Lee and Deane, 2009) to a full 3D MHD scheme. The scheme is a finite-volume Godunov method consisting of a constrained transport (CT) method and an efficient and accurate single-step, directionally unsplit multidimensional data reconstruction-evolution algorithm, which extends the original 2D corner transport upwind (CTU) method (Colella, 1990). We present two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a simple 3D extension of the 2D CTU method by Colella (1990) and is considered as a direct extension from the 2D USM scheme. The full 3D CTU scheme is our primary 3D solver which includes all multidimensional cross-derivative terms for stability. The latter method is logically analogous to the 3D unsplit CTU method by Saltzman. The major novelties in our algorithms are twofold. First, we extend the reduced CTU scheme to the full CTU scheme which is able to run with CFL numbers close to unity. Both methods utilize the transverse update technique developed in the 2D USM algorithm to account for transverse fluxes without solving intermediate Riemann problems, which in turn gives cost-effective 3D methods by reducing the total number of Riemann solves. The proposed algorithms are simple and efficient especially when including multidimensional MHD terms that maintain in-plane magnetic field dynamics. Second, we introduce a new CT scheme that makes use of proper upwind information in taking averages of electric fields.