Multidimensional HLLI generalized riemann problem solver for conservation laws – The two-dimensional case for structured meshes

The Riemann problem, and the associated generalized Riemann problem (GRP), are increasingly seen as important building blocks for modern higher order Godunov-type schemes. While most solutions of the GRP are specific to a particular hyperbolic law, a general-purpose GRP that can be applied to any hyperbolic conservation law has emerged in the form of the one-dimensional HLLI-GRP. Approximate multidimensional Riemann solvers have also been designed by the first author and his colleagues. However, a multidimensional GRP that is applicable to any hyperbolic conservation law has never been designed until now to the best of our knowledge. It this paper, we accomplish such a task.

The study of the multidimensional Riemann problem entails the study of the strongly-interacting state. Starting with the multidimensional HLL-based Riemann solver, we present all the steps for endowing spatial gradients to the strongly-interacting state. This is accomplished through application of Rankine-Hugoniot shock jump conditions to the higher order terms in a Taylor series expansion of the strongly-interacting state. A linearized formulation is also used to obtain the spatial gradients to the strongly-interacting state. With the spatial gradients in hand, it is possible to specify the multidimensional HLL flux as well its time-derivative. This results in a multidimensional HLL-GRP solver. We then utilize intermediate waves to reduce the dissipation of the multidimensional HLL-GRP. This gives us an HLLI-GRP solver which significantly reduces dissipation and is complete in multiple dimensions.