Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
Notes on the eigensystem of magnetohydrodynamics
SIAM Journal on Applied Mathematics
Wave propagation algorithms for multidimensional hyperbolic systems
Journal of Computational Physics
Journal of Computational Physics
A simple finite difference scheme for multidimensional magnetohydrodynamical equations
Journal of Computational Physics
Journal of Computational Physics
A solution-adaptive upwind scheme for ideal magnetohydrodynamics
Journal of Computational Physics
The &Dgr; • = 0 constraint in shock-capturing magnetohydrodynamics codes
Journal of Computational Physics
A wave propagation method for three-dimensional hyperbolic conservation laws
Journal of Computational Physics
Hyperbolic divergence cleaning for the MHD equations
Journal of Computational Physics
SIAM Journal on Scientific Computing
Conservative and orthogonal discretization for the Lorentz force
Journal of Computational Physics
AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension)
AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension)
Journal of Computational Physics
An unsplit Godunov method for ideal MHD via constrained transport
Journal of Computational Physics
An Unstaggered, High-Resolution Constrained Transport Method for Magnetohydrodynamic Flows
SIAM Journal on Scientific Computing
An unsplit Godunov method for ideal MHD via constrained transport in three dimensions
Journal of Computational Physics
E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme
Journal of Computational Physics
Hi-index | 31.45 |
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 [J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput. 28 (2006) 1766], and is based on the high-resolution wave propagation method of Langseth and LeVeque [J.O. Langseth, R.J. LeVeque, A wave propagation method for threedimensional hyperbolic conservation laws, J. Comput. Phys. 165 (2000) 126]. 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.