Abstract
An approximate Riemann solver is developed for the governing equations of ideal magnetohydrodynamics (MHD). The Riemann solver has an eight-wave structure, where seven of the waves are those used in previous work on upwind schemes for MHD, and the eighth wave is related to the divergence of the magnetic field. The structure of the eighth wave is not immediately obvious from the governing equations as they are usually written, but arises from a modification of the equations that is presented in this paper. The addition of the eighth wave allows multi-dimensional MHD problems to be solved without the use of staggered grids or a projection scheme, one or the other of which was necessary in previous work on upwind schemes for MHD. A test problem made up of a shock tube with rotated initial conditions is solved to show that the two-dimensional code yields answers consistent with the one-dimensional methods developed previously.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. U. Brackbill and D. C. Barnes. The effect of nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations. Journal of Computational Physics, 35: 426 – 430, 1980.
M. Brio and C. C. Wu. An upwind differencing scheme for the equations of ideal magnetohydrodynamics. Journal of Computational Physics, 75: 400 – 422, 1988.
A. Jeffrey and A. Taniuti. Nonlinear Wave Propagation. Academic Press, 1964.
P. L. Roe, 1994. Private Communication.
P. L. Roe and D. S. Balsara. Notes on the eigensystem of magnetohydrodynamics. Journal of Computational Physics., 1994. To appear.
G. A. Sod. A survey of finite-difference methods for systems of nonlinear conservation laws. Journal of Computational Physics, 27 (1): 1 – 31, 1978.
P. R. Woodward and W. Dai. A Riemann solver for ideal magnetohydrodynamics. Journal of Computational Physics. 1994. To appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Powell, K.G. (1997). An Approximate Riemann Solver for Magnetohydrodynamics. In: Hussaini, M.Y., van Leer, B., Van Rosendale, J. (eds) Upwind and High-Resolution Schemes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60543-7_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-60543-7_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64452-8
Online ISBN: 978-3-642-60543-7
eBook Packages: Springer Book Archive