A two-dimensional Riemann solver with self-similar sub-structure – Alternative formulation based on least squares projection

Dinshaw S. Balsara (1) , Jeaniffer Vides (2, 3) , Katharine Gurski (4) , Boniface Nkonga (5, 3) , Michael Dumbser (6) , Sudip Garain (1) , Edouard Audit (2)

1 Department of Physics [Notre Dame]
2 MDLS - Maison de la Simulation
3 CASTOR - Control, Analysis and Simulations for TOkamak Research
4 Howard - Department of Mathematics
5 LJAD - Laboratoire Jean Alexandre Dieudonné
6 Department of civil, environmental and mechanical engineering [Trento]

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 together 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.

Mots clés: Multidimensional Riemann solvers; Conservation laws; Hyperbolic systems; Higher order Godunov schemes

Domaines: Dynamique des Fluides [physics.flu-dyn]; Analyse numérique [math.NA]; Physique des plasmas [physics.plasm-ph]

Dinshaw S. Balsara, Jeaniffer Vides, Katharine Gurski, Boniface Nkonga, Michael Dumbser, et al.. A two-dimensional Riemann solver with self-similar sub-structure – Alternative formulation based on least squares projection. Journal of Computational Physics, 2016, 304, pp.138-161. ⟨10.1016/j.jcp.2015.10.013⟩. ⟨hal-01254231⟩