Exact solution for Riemann problems of the shear shallow water model

Abstract

The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.

A new class of sediment transport models: Derivation and numerical solutions

“…Our scheme accurately resolves the values of h, Z b , and P. The plotted profile of h shows that the waves used in the PCCU scheme are identical to those used in the HLL scheme, and both methods accurately capture all internal waves. The same results are also obtained in [31] and [8] using 3-waves and 4-waves approximate Riemann solvers. The test prove that our scheme is efficient…”

“…This test is designed to verify the accuracy of the proposed scheme. Similar tests are performed with the same parameters in [8], [31] with similar methods. In this example, a reduced model (distortion two-velocity shallow water model) obtained by removing all the source terms is performed and analyzed.…”

“…This problem fails some representation test cases in recent work of [8], [31] and [28]. We will attempt to design shock-capturing the path-conservative scheme by using a great number of intermediate waves between two states as in [8] and [31]. However, it is observed in a recent work that the number of waves does not have any influence on the convergence of the scheme.…”

“…This problem fails some representation test cases in recent work of [8], [31] and [28]. We will attempt to design shock-capturing the path-conservative scheme by using a great number of intermediate waves between two states as in [8] and [31].…”