The Riemann problem for the shallow water equations with discontinuous topography

Abstract

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.

A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime

“…There are several improvements in the construction of Riemann solutions in this paper over the ones in our previous work [33]. First, we can determine larger domains of existence by combining constructions in [33] together. Second, the domains where there is a unique solution or there are several solutions are precisely determined.…”

“…As discussed in [33], across a discontinuity there are two possibilities: (i) either the bottom height a remains constant,…”

“…Recall that, in LeFloch and Thanh [33], a first investigation of the Riemann problem for the shallow water equations was performed. The present paper provides a very significant improvement in that a complete description of a Riemann solver is now obtained for the first time.…”

The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section

“…In consequence, at least within the regime where the system is strictly hyperbolic, the theory of such systems developed by LeFloch and co-authors (see [7] and also [16][17][18][19][20][21][22]) applies, and provide the existence of entropy solutions to the Riemann problem (a single discontinuity separating two constant states as an initial data), as well as to the Cauchy problem (for solution with sufficiently small total variation). More recently, LeFloch and Thanh [21,22] solved the Riemann problem for arbitrary data, including the regime where the system fails to be globally strict hyperbolicity (i.e., the resonant case).…”

“…More recently, LeFloch and Thanh [21,22] solved the Riemann problem for arbitrary data, including the regime where the system fails to be globally strict hyperbolicity (i.e., the resonant case). The Riemann problem was also solved by a different approach by Andrianov and Warnecke [1].…”