Investigation of Riemann Solver with Internal Reconstruction (RSIR) for the Euler Equations (Communications in Computational Physics, 2021)
The Riemann solver with internal reconstruction (RSIR) of Carmouze et al.(2020) is investigated, revisited and improved for the Euler equations. In this reference,the RSIR approach has been developed to address the numerical resolution of the non-equilibrium two-phase flow model of Saurel et al. (2017). The main idea is to reconstruct two intermediate states from the knowledge of a simple and robust intercell statesuch as HLL, regardless the number of waves present in the Riemann problem. Suchreconstruction improves significantly the accuracy of the HLL solution, preserves robustness and maintains stationary discontinuities. Consequently, when dealing withcomplex flow models, such as the aforementioned one, RSIR-type solvers are quiteeasy to derive compared to HLLC-type ones that may require a tedious analysis of thegoverning equations across the different waves. In the present contribution, the RSIRsolver of Carmouze et al. (2020) is investigated, revisited and improved with the helpof thermodynamic considerations, making a simple, accurate, robust and positive Riemann solver. It is also demonstrated that the RSIR solver is strictly equivalent to theHLLC solver of Toro et al. (1994) for the Euler equations when the Rankine-Hugoniotrelations are used. In that sense, the RSIR approach recovers the HLLC solver in somelimit as well as the HLL one in another limit and can be simplified at different levels when complex systems of equations are addressed. For the sake of clarity andsimplicity, the derivations are performed in the context of the one-dimensional Eulerequations. Comparisons and validations against the conventional HLLC solver andexact solutions are presented.
Keywords: Riemann solver; RSIR; HLL; HLLC; hyperbolic.