An Unsplit Godunov Method for Ideal MHD via Constrained Transport in Three Dimensions

TL;DR

This paper introduces a second-order accurate, single-step Godunov scheme for ideal MHD in three dimensions that combines the CTU method with constrained transport to maintain divergence-free magnetic fields, demonstrating high accuracy and robustness.

Contribution

It extends a 2D Godunov scheme to 3D for ideal MHD, integrating CTU and constrained transport methods to ensure divergence-free magnetic fields in a single step.

Findings

The scheme accurately captures MHD phenomena in 3D test problems.

The method maintains the divergence-free condition of magnetic fields.

It demonstrates robustness across various test scenarios.

Abstract

We present a single step, second-order accurate Godunov scheme for ideal MHD which is an extension of the method described by Gardiner & Stone (2005) to three dimensions. This algorithm combines the corner transport upwind (CTU) method of Colella for multidimensional integration, and the constrained transport (CT) algorithm for preserving the divergence-free constraint on the magnetic field. We describe the calculation of the PPM interface states for 3D ideal MHD which must include multidimensional ``MHD source terms'' and naturally respect the balance implicit in these terms by the ${\bf\nabla\cdot B}=0$ condition. We compare two different forms for the CTU integration algorithm which require either 6- or 12-solutions of the Riemann problem per cell per time-step, and present a detailed description of the 6-solve algorithm. Finally, we present solutions for test problems to demonstrate the accuracy and robustness of the algorithm.