Genuine Multidimensional, Divergence-free Numerical Method for the Equations of Magnetohydrodynamics

Description

The equations of magnetohydrodynamics (MHD) describe the flow of plasmas in interaction with a magnetic field. MHD equations are relevant in investigations in several areas of engineering and astrophysical research.

The MHD equations form a system of hyperbolic partial differential equations, which will be solved in this dissertation project by use of the numerical scheme 'Method of Transport' (MoT). The Method of Transport was developed as genuine multidimensional numerical scheme for the Euler equations (M. Fey). At the Seminar for Applied Mathematics the ideas of MoT have been adopted to other systems, for example the shallow water equations (A.T. Morel) or turbulent flow (J. Maurer).
In MHD two new difficulties arise compared to the Euler or shallow water equations. First, the characteristic surfaces are no longer spheres. The influence of the magnetic field reduces the symmetry of the problem at least by one space dimension. Second, the divergence of the magnetic field has to be zero at any time, i.e. an additional constraint has to be imposed on the solution in each time step. Fortunately, these constraint is inherent to the equations, i.e. once fulfilled at the initial data it is fulfilled for all times. Thus this constraint does not change the character of the equations, as in the case of the incompressible Euler equations.

It could be shown, that the additional constraint may be incorporated into the numerical method by special 2-dimensional-flux formulations within the framework of MoT (R. Limacher). This approach will be theoretically justified and extended to three dimensions.The large computations are parallelized and performed at the Beowulf cluster 'Asgard' at ETH Zürich.

Contacts

Prof. Rolf Jeltsch and Manuel Torrilhon.