Genuine Multidimensional Discretization of non linear System of conservation Laws

Description

The development of the Method of Transport for the Euler equations (MoT) makes extensive use of the homogeneity of the flux. A.T. Morel was able to adapted this idea to the shallow water equations, a system with inhomogeneous flux. In the future we will address other systems, too. Especially the equations of magneto hydrodynamics (MHD). Two new difficulties arise compared to the Euler- and 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, an additional constrain has to be imposed on the solution in each time step. Fortunately, these constrains are inherent to the equations, i.e. once full filed at the initial data they are fulfilled for all times. Thus this constrain does not change the character of the equations, as in the case of the incompressible Euler equations.

Modification to the basic ideas of the MoT are planed to improve the numerical properties of the method in view of treating diffusive effects (see LES and Combustion, J. Maurer), better approximation of steady solutions (A.T. Morel), truly multidimen- sional discretization of divergence free conditions as in the case of incompressible flow or MHD, high boundary conditions for cartesian grids in complex domains (H. Forrer) or adaptive mesh coarsening to reduce the computational costs.

Contacts

Prof. Rolf Jeltsch and Michael Fey (fey@sam.math.ethz.ch).