We investigate the Riemann problem for the shallow water equations with variable and (possibly) discontinuous topography and provide a complete description of the properties of its solutions: existence; uniqueness in the non-resonant regime; multiple solutions in the resonant regime. This analysis leads us to a numerical algorithm that provides one with a Riemann solver. Next, we introduce a Godunov-type scheme based on this Riemann solver, which is well-balanced and of quasi-conservative form. Finally, we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime, except in the limiting situation when Riemann data precisely belong to the resonance hypersurface.
Abstract. We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is non-strictly hyperbolic and does not admit a fully conservative form, and we establish the existence of two-parameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit, and leads to formulas that can be implemented numerically for the approximation of the general initial-value problem.
We consider a nonlinear hyperbolic model describing phase transitions in nonlinear elastodynamics. The Riemann problem is solved uniquely, provided we supplement the fundamental conservation laws (mass, momentum) with a kinetic relation. The latter takes into account small-scale mechanisms, such as the viscosity and capillary effects in the material under consideration. Our construction generalizes, to solutions of arbitrary large amplitude of the model under study, an approach proposed by Hayes and LeFloch for general systems of conservation laws. The Riemann solutions may contain rarefaction waves, (compressive) classical shock waves, as well as (undercompressive) nonclassical shocks.Mathematics Subject Classification (1991). Primary: 35L65, 74XX. Secondary: 76N10, 76L05.
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