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

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

“…We refer the reader to [1,8,35] for partial or alternative approaches to the Riemann problem. On the other hand, Chinnayya, LeRoux, and Seguin [11] introduced a Godunov method for (1.1) based on a Riemann solver determined by "continuation": they start their construction by assuming that the bottom topography is flat and then extend it to a non-flat topography.…”

Two-dimensional simulation of debris flows over mobile bed: Enhancing the TRENT2D model by using a well-balanced Generalized Roe-type solver

“…This integral cannot be easily expressed in a general form, as it depends on the pressure distribution, which is a function of the bed step and on the velocity field. Lacking experimental evidences, the following relation, proposed by [30] (used also in [31,21] for the mobile-bed case and in [4,32] for the fixed bed case), will be used here:…”

“…If unphysical fluxes are employed in updating a cell, the system that allow the passage from conserved variables to primitive ones (see Eq. (32) and (33)) have no solution. The present model has been used in a very large number of applications of any kind but this problem has never appeared.…”

A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section

“…The reader is referred to [23,20] for the Riemann problem for the isentropic model (1.1), to [29] for the Riemann problem for the model of a general fluid in a nozzle with discontinuous cross-section, to [21,22,7,25] for the Riemann problem for the shallow water equations with discontinuous topography, to [28,26] for the Riemann problem for two-phase flow models, and to [16,17,12] for the Riemann problem for other hyperbolic nonconservative models. See [10] for the standard Godunov scheme of systems of conservation laws.…”