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

High-order conservative finite difference GLM-MHD schemes for cell-centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175 (2002) 645-673). The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics (Aug 7 2009

Positivity-Preserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations

In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high order weighted essentially non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under the constrained transport (CT) framework, can achieve high order accuracy, a discrete divergence-free condition and positivity of the numerical solution simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate the performance of the proposed method.Comment: 21 pages, 28 figure