TY - JOUR
T1 - An unsplit Godunov method for ideal MHD via constrained transport
AU - Gardiner, Thomas A.
AU - Stone, James McLellan
N1 - Funding Information:
The authors thank Charles Gammie, John Hawley, and Eve Ostriker for discussion and comments on an early draft of this paper. We thank Peter J. Teuben for contributions to the implementation of the algorithms described here. This work was supported by the National Science Foundation ITR Grant AST-0413788. J.S. thanks the University of Cambridge and the Royal Society for financial support during the course of this work.
PY - 2005/5/20
Y1 - 2005/5/20
N2 - We describe a single step, second-order accurate Godunov scheme for ideal MHD based on combining the piecewise parabolic method (PPM) for performing spatial reconstruction, 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 adopt the most compact form of CT, which requires the field be represented by area-averages at cell faces. We demonstrate that the fluxes of the area-averaged field used by CT can be made consistent with the fluxes of the volume-averaged field returned by a Riemann solver if they obey certain simple relationships. We use these relationships to derive new algorithms for constructing the CT fluxes at grid cell corners which reduce exactly to the equivalent one-dimensional solver for plane-parallel, grid-aligned flow. We show that the PPM reconstruction algorithm must include multidimensional terms for MHD, and we describe a number of important extensions that must be made to CTU in order for it to be used for MHD with CT. We present the results of a variety of test problems to demonstrate the method is accurate and robust.
AB - We describe a single step, second-order accurate Godunov scheme for ideal MHD based on combining the piecewise parabolic method (PPM) for performing spatial reconstruction, 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 adopt the most compact form of CT, which requires the field be represented by area-averages at cell faces. We demonstrate that the fluxes of the area-averaged field used by CT can be made consistent with the fluxes of the volume-averaged field returned by a Riemann solver if they obey certain simple relationships. We use these relationships to derive new algorithms for constructing the CT fluxes at grid cell corners which reduce exactly to the equivalent one-dimensional solver for plane-parallel, grid-aligned flow. We show that the PPM reconstruction algorithm must include multidimensional terms for MHD, and we describe a number of important extensions that must be made to CTU in order for it to be used for MHD with CT. We present the results of a variety of test problems to demonstrate the method is accurate and robust.
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U2 - 10.1016/j.jcp.2004.11.016
DO - 10.1016/j.jcp.2004.11.016
M3 - Article
AN - SCOPUS:17144378992
SN - 0021-9991
VL - 205
SP - 509
EP - 539
JO - Journal of Computational Physics
JF - Journal of Computational Physics
IS - 2
ER -