Numerical MHD is a PDE that is prototypical of an involution-constrained system. Two new concepts get introduced here. First, magnetic fields evolve in a divergence-free fashion and we introduce the concept of divergence-free reconstruction. Second, the electric fields that are needed at edges and require multidimensional upwinding. Such multidimensional upwinding can only be provided via a genuinely multidimensional Riemann solver. We present both these concepts here.

Multidimensional Riemann solvers can be formulated in a couple of different ways. There is the space-time formulation which is based on taking a geometrical viewpoint. It can be modified to represent sub-structures, like contact discontinuities, in the multidimensional Riemann solver. The alternative approach consists of formulating the problem in similarity variables. The latter formulation yields a Galerkin-like approach and can also be used to endow sub-structure to the strongly interacting state. In this Appendix, we provide information on both formulations.

The eventual intention is to provide an entire chapter on this and allied topics. But, the ideas contained here are high-interest. As a result, this place-holder Appendix is provided which explains some of these novel concepts using talks that the author has given.

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