Shear shallow water model : Energy consistent schemes.
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We propose a complete study of the Riemann problem for a genuinely non-conservative hyperbolic system derived from modeling reasonably turbulent thin flows. This model coincides with the ten-moment model derived from gas kinetics in the presence of a gravi- tational potential. Although the model is fundamentally non-conservative, the generalized jump relations should be compatible with the conservation of the total energy. Conse- quently, the generalized jump relations consistent with energy conservation are indepen- dent of the path chosen. In this context, we show that the exact solution of the Riemann problem exists and is unique, except in the case of vacuum appearance. This exact Riemann solver can work well within Glim and Godunov’s numerical framework but at a prohibitive computational cost. We also propose approximate solvers with two, three, or five waves of discontinuities: HLL, HLLC3, and HLLC5 .
