This paper investigates the origin of divergence errors in MHD simulations. For that
purpose, we introduce the concept of discrete involutions for discretized
conservation laws. This is done in analogue to the concept of involutions for
hyperbolic conservation laws, introduced by Dafermos. By exploring the
connection between discrete involutions and resonance, especially for constrained
transport like MHD, we identify the lack of positive central viscosity and the
assumption of one-dimensional physics in the calculation of intercell fluxes as
the main sources of divergence errors. As an example of the consequences
for numerical schemes, we give a hint how to modify Roe-type schemes in
order to decrease the divergence errors considerably and, thus, stabilize the
scheme.
Institute for Applied Mathematics
and Scientific Computing
Brandenburg University of Technology Cottbus
Platz der Deutschen Einheit 1
D-03046 Cottbus
Germany
