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Divergence Constraint
in cooperation with Friedemann Kemm and Claus-Dieter Munz
In numerical simulations the divergence constraint on the magnetic field causes severe stability problems. Accumulating errors can lead to an unphysical situation and can result in a breakdown of the simulation. We have developed an approach to the stabilization of numerical schemes which can be easily used as an extension of an existing solver. The method is based on a modified formulation of the MHD equations in which the divergence constraint is coupled to the system by introducing a further unknown function. The evolution of divergence errors is strongly dependent on the type of the equation chosen for this function. For the one-dimensional setting we show that these errors can be transported out of the computational domain by a wave equation or can be dissipated by a heat equation. We suggest a mixed formulation by which the divergence errors are transported and dissipated at the same time. The resulting system is still hyperbolic and the density, momentum, magnetic induction, and the total energy density are still conserved. In numerical examples we have seen that our method decreases the divergence errors by up to two orders of magnitude even compared with the often used source term stabilization approach by Powell et al.
Results of numerical tests comparing several divergence cleaning methods by means of the 2D Riemann problem.
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A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, and M. Wesenberg, Hyperbolic Divergence Cleaning for the MHD Equations. J. Comput. Phys. 175, pp. 645-673 (2002).
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A. Dedner, D. Kröner, C. Rohde, and M. Wesenberg, Efficient Divergence Cleaning in Three-Dimensional MHD Simulations. In: High Performance Computing in Science and Engineering '02, edited by E. Krause and W. Jäger, pp. 323-334 (Springer, 2003).
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A. Dedner, C. Rohde, and M. Wesenberg, A New Approach to Divergence Cleaning in Magnetohydrodynamic Simulations. To appear in the proceedings of the 9th international conference on Hyperbolic Problems: Theory, Numerics, Applications, Pasadena, CA, USA (2002).
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