An Unstaggered Constrained Transport Method for the 3D Ideal Magnetohydrodynamic Equations

Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must either confront the challenge of controlling errors in the discrete divergence of the magnetic field, or else be faced with nonlinear numerical instabilities. One approach for controlling the discrete divergence is through a so-called constrained transport method, which is based on first predicting a magnetic field through a standard finite volume solver, and then correcting this field through the appropriate use of a magnetic vector potential. In this work we develop a constrained transport method for the 3D ideal MHD equations that is based on a high-resolution wave propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme developed by Rossmanith [SIAM J. Sci. Comp. 28, 1766 (2006)], and is based on the high-resolution wave propagation method of Langseth and LeVeque [J. Comp. Phys. 165, 126 (2000)]. In particular, in our extension we take great care to maintain the three most important properties of the 2D scheme: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as cell-centered; (2) we develop a high-resolution wave propagation scheme for evolving the magnetic potential; and (3) we develop a wave limiting approach that is applied during the vector potential evolution, which controls unphysical oscillations in the magnetic field. One of the key numerical difficulties that is novel to 3D is that the transport equation that must be solved for the magnetic vector potential is only weakly hyperbolic. In presenting our numerical algorithm we describe how to numerically handle this problem of weak hyperbolicity, as well as how to choose an appropriate gauge condition. The resulting scheme is applied to several numerical test cases.Comment: 46 pages, 12 figure

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

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.Comment: Extended version of the paper accepted for publication in JC