Summary
We propose a procedure to initialize the magnetic flux, possibly discontinuous, on a finite volume grid in an exactly solenoidal way. That is, a certain discrete divergence operator will vanish on each cell. Combined with a new locally divergence preserving numerical scheme we are able to conduct MHD simulations which have an exactly vanishing discrete divergence. In this paper we describe the new scheme and the initialization procedure and present the results of a simulation of a shock interaction with a magnetized cloud.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Balsara, D. S. and Spicer, D. S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comp. Phys. 149, (1999) p.270
Brackhill, J. U. and Barnes, D. C., The effect of nonzero Δ · B on the numerical solution of the magnetohydrodynamic equations, J. Comp. Phys. 35, (1980) p.426
Dai, W. and Woodward, P. R., On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows, Astrophys. J. 494, (1998) p.317
Dedner, A., Kemm, F., Kröner, D., Munz, C.-D., Schnitzer, T., and Wesenberg, M., Hyperbolic Divergence Cleaning for the MHD Equations, J. Comp. Phys. 175(2), (2002) p.645
DeSterck, H., Multi-Dimensional Upwind Constrained Transport on Unstructured Grids for Shallow Water Magnetohydrodynamics, AIAA Paper 2001–2623, (2001)
Evans, C. R. and Hawley, J. F., Simulation of Magnetohydrodynamic Flows: A Constrained Transport Method, Astrophys. J. 332, (1988) p.659
Godlewski, E. and Raviart, P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, New York (1996)
Jeffrey, A. and Taniuti, T., Non-linear Wave Propagation, Academy Press, New York (1964)
Jiang, G.-S. and Shu, C.-W., Efficient Implementation of weighted ENO schemes, J. Comp. Phys. 126, (1996) p.202
Munz, C.-D., Omnes, P., Schneider, R., Sonnendrücker, E., and Voss, U., Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model, J. Comp. Phys. 161(2), (2000), p.484
Powell, K. G., An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), ICASE Report No. 94-24, (1994)
K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi, and D. L. DeZeeuw, A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comp. Phys. 154/2 (1999) p.284
Torrilhon, M. and Fey, M., Multidimensional Upwind Methods for Advection Equations with Constraints, submitted to SIAM J. Num. Anal., (2003)
Torrilhon, M., Locally Divergence-preserving Upwind Schemes for Magnetohydrodynamic Equations, submitted to SIAM J. Sci. Comp., (2003)
Toth, G., The Δ · B Constraint in Shock-Capturing Magnetohydrodynamics Codes, J. Comp. Phys. 161, (2000)
M. Wesenberg, Efficient MHD Riemann Solvers for Simulations on Unstructured Triangular Grids, East West J. Numer. Math. in press (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jeltsch, R., Torrilhon, M. (2005). Solenoidal Discrete Initialization for Magnetohydrodynamics. In: Bock, H.G., Phu, H.X., Kostina, E., Rannacher, R. (eds) Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27170-8_19
Download citation
DOI: https://doi.org/10.1007/3-540-27170-8_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23027-4
Online ISBN: 978-3-540-27170-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)