Numerical Treatment of Wet/Dry Fronts in Shallow Flows With a Modified Roe Scheme

Abstract

This paper deals with the analysis of some numerical difficulties related to the appearance of wet/dry fronts that may occur during the simulation of free-surface waves in shallow fluids. The fluid is supposed to be governed by the Shallow Water equations and the discretization of the equations is performed, when wet/dry fronts do not appear, by means of the Q-scheme of Roe upwinding the source terms introduced in Ref. 40. This scheme is well-balanced in the sense that it solves exactly stationary solutions corresponding to water at rest. Wet/dry fronts cannot be correctly treated with this scheme: it can produce negative values of the thickness of the fluid layer and stationary solutions corresponding to water at rest including wet/dry transitions are not exactly solved. In Refs. 3–5 some variants of this numerical scheme have been proposed that partially solve these difficulties. Here we propose a new variant: at intercells where wet/dry transitions occur, a Nonlinear Riemann Problem is considered instead of a Linear one. The exact solutions of these nonlinear problems, which are easy to calculate, are used in order to define the numerical fluxes. We investigate the properties of the resulting scheme and present some comparisons with the numerical results obtained with some other modified numerical schemes proposed previously.

Performance Benchmarking of Tsunami-HySEA Model for NTHMP’s Inundation Mapping Activities

“…It has been thoroughly tested, and in particular has passed not only all tests by Synolakis et al (2008), but also other laboratory tests and proposed benchmark problems. Some of them can be found in the studies by Castro et al (2005Castro et al ( , 2006, Gallardo et al (2007), de la , and NTHMP (2016).…”

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

“…Note also that we have checked that the entropy fix correction implemented in our relaxation solver does not intervene in the computation of this particular experiment, where in fact there are no problems of entropy violating solutions associated to transonic rarefactions. Let us also mention that in [25] the test problem presented here is solved by a modified Roe method [24,25], called MRoe, which is not rigorously positivity preserving. The authors need to reduce the CFL number to 0.8 in order to avoid the appearance of negative values of the water height.…”

“…Other approximate solvers able to treat efficiently vacuum states have been developed by means of relaxation strategies, such as Suliciu's solver [18][19][20], and the recent method of Berthon-Marche [21]. Among other methods, let us mention the augmented four-wave Riemann solver of George [22], which is related to the class of relaxation solvers of LeVeque and Pelanti [23], and the modified Roe method (MRoe) of Castro, Parés and co-workers [24,25]. This MRoe method however is not rigorously positivity preserving, and it may need a restriction on the CFL number to avoid negative water depths.…”

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“…Indeed, they may produce nonphysical negative values of the thickness of the water layer near the wet/dry front. Some ways to modify Roe's method to fix these problems have been proposed in [11,10].…”