High Order Local Divergence Preservation for MHD Finite Volume Methods

Violation of the divergence constraint on the magnetic ux density in magnetohydrodynamical (MHD) simulations leads to stability problems and it is therefore of great importance to numerically respect this intrinsic constraint. Especially in shock computations with discontinous elds, divergence errors tend to accumulate and introduce strong oscillations. In [1] it has been shown that an intrinsic dierential constraint can be incorporated into the ux of a nite volume method by respecting special shape functions as ux distribution. This leads to a seamless and local divergence preservation. The second order scheme for MHD has been investigated in [2]. This paper presents a higher order extension of the Figure 1: fourth and second order divergence preserving ux distributions for the magnetic eld locally divergence-preserving procedure developed in [2]. Therefore the theory of constraint preserving ux distribution is utilized on the basis of a fourth order divergence operator. The new divergence preserving shape functions for uxes are shown in Fig. 1 in com-parison to the second order case. In a second step these ux distributions are combined such that consistency with the evolution equation is of fourth order as well, see [3]. The redistribution ensures that a fourth order accurate discrete divergence operator is preserved to round o errors when applied to the cell averages of the magnetic ux density. The developed procedure can be combined to arbitrary semi-discrete nite volume schemes and its purpose is to stabilize the schemes using a local procedure that respects the accuracy of the base scheme to a greater extent than the previous second order achievements. Theoretical justication and numerical experiments that demonstrate the properties of the new procedure will be presented. References: [1] Torrilhon M. and Fey M., Constraint-preserving upwind methods for multidimensional advection equations