Just as
the quality of a one-dimensional approximate Riemann solver is improved by
the inclusion of internal sub-structure,
the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come toge
ther at
the vertex of a mesh. The interaction of
the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The self-similar formulation of Balsara
[16] proves especially useful for this purpose. While that work is based on a Galerkin projection, in this paper we present an analogous self-similar formulation that is based on a different interpretation. In
the present formulation, we interpret
the shock jumps at
the boundary of
the strongly-interacting state quite literally. The enforcement of
the shock jump conditions is done with a least squares projection (Vides, Nkonga and Audit
[67]). With that interpretation, we again show that
the multidimensional Riemann solver can be endowed with sub-structure. However, we find that
the most efficient implementation arises when we use a flux vector splitting and a least squares projection. An alternative formulation that is based on
the full characteristic matrices is also presented. The multidimensional Riemann solvers that are demonstrated here use one-dimensional HLLC Riemann solvers as building blocks.
Several stringent test problems drawn from hydrodynamics and MHD are presented to show that the method works. Results from structured and unstructured meshes demonstrate the versatility of our method. The reader is also invited to watch a video introduction to multidimensional Riemann solvers on class="interref" data-locatorType="url" data-locatorKey="http://www.nd.edu/~dbalsara/Numerical-PDE-Course">http://www.nd.edu/~dbalsara/Numerical-PDE-Course.