Performance of numerical methods on the non‐unique solution to the Riemann problem for the shallow water equations

Abstract

SUMMARYFor certain initial conditions, the exact solution to the Riemann problem for the shallow water equations is not unique. We test the performance of several numerical methods on such initial data and establish that the numerical solution can pick out di erent exact solutions. Moreover, the numerical solution does not necessarily converge towards the picked-out exact solution.

An exact Riemann solver for compressible two-phase flow models containing non-conservative products

“…Dal Maso et al [10], Lefloch and Tzavaras [11] and Crasta and Lefloch [12]), an existential theory for such equations is not yet available. More important, it has been demonstrated that due to the non-conservative products, the balance equations admit more than an unique solution under certain flow conditions, Andrianov and Warnecke [13] and Andrianov [29].…”

“…Non-uniqueness of solutions has also been encountered in the compressible, quasi-1D duct flow equations [13,28], in the shallow-water equations [29], and elsewhere. In sofar, various propositions have been put forward to select the physically relevant, ''correct'', solution when multiple solutions are present.…”

A well-balanced approach for flows over mobile-bed with high sediment-transport

“…The relevant exact solutions have been worked out by a process of subsequent wave-field construction, starting from either initial-state (left or right), and coming to define, in the end, the other one. In so doing an inverse problem is solved in the way outlined by [2,13].…”

The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations

“…For a number of step Riemann Problems, numerical predictions obtained by using the proposed approach have been compared to the exact solutions, obtained by an inverse procedure [2,17], achieving good results.…”

“…However, in both works there is awareness that dissipation actually occurs in the recirculation cell located at the inner corner of the step and therefore they introduce the possibility of a loss in the energy relation. Galloüet et al [7], Chinnayya et al [6], and Andrianov [2] claim that, since the standing wave over the step is a contact discontinuity, it must be characterized by the constancy of the relevant Riemann Invariants (RIs), namely mass and energy, in any situation except in the resonant case. This result is rather controversial because from the mathematical point of view energy losses are not admitted, while from the physical point of view it is clear that they actually exist.…”