A well-balanced scheme for the shallow-water equations with topography

Abstract

International audienceA non-negativity preserving and well-balanced scheme that exactly preserves all the smooth steady states of the shallow water system, including the moving ones, is proposed. In addition, the scheme must deal with vanishing water heights and transitions between wet and dry areas. A Godunov-type method is derived by using a relevant average of the source terms within the scheme, in order to enforce the required well-balance property. A second-order well-balanced MUSCL extension is also designed. Numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states

A Second-Order Well-Balanced Scheme for the Shallow Water Equations with Topography

“…We consider the well-balanced numerical scheme for the shallow-water equations with topography introduced in [8] and its second-order well-balanced extension, which requires two heuristic parameters. The goal of the present contribution is to derive a parameter-free second-order well-balanced scheme.…”

“…schemes that exactly preserve such steady solutions, have been derived in the last decade (see for instance [2,4,5,8]). Namely, in [8], the authors suggested a well-balanced Godunov-type scheme based on a two-state approximate Riemann solver. We briefly recall the general form of a numerical scheme that falls within this classification (see for instance [10]).…”

“…To address this issue, an approximate Riemann solver is introduced. More specifically, in [8], the authors develop an approximate Riemann solver satisfying several crucial properties: consistency, well-balance and preservation of the water height non-negativity. The time update of the approximate solution in the cell x i reads:…”

“…However, the downside of this approach is that, in each cell, the nonlinear steady relations (2) have to be solved, thus leading to extra computational cost. In [8], the authors proposed a convex combination, in each cell c i and at time t n+1 , between the well-balanced scheme and a MUSCL reconstruction to recover the wellbalance property without having to solve nonlinear equations, as follows:…”

“…where W L, * [8] prove that the Godunov-type scheme (3) is consistent, well-balanced and non-negativity-preserving.…”