On some fast well-balanced first order solvers for nonconservative systems

Abstract

Abstract. The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good wellbalanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

“…The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see [14]). This Riemann solver is based on a suitable decomposition of a Roe matrix (see [27]) by means of a parabolic viscosity matrix (see [16]) that captures some information concerning the intermediate characteristic fields.…”

“…Each layer is assumed to have a constant density, ρ i , i = 1, 2 (ρ 1 < ρ 2 ). The unknowns q i (x, t) and h i (x, t) represent respectively the mass-flow and the thickness of the ith layer at the section of coordinate x at time t. The numerical resolution of two-layer or multilayer shallow water systems has been object of an intense research during the last years: see for instance [1], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [17], [22], [24] . .…”

Efficient simulation of one-dimensional two-phase flow with a high-order h-adaptive space-time Discontinuous Galerkin method

“…See for example Chalmers and Lorin [23] for a discussion on choosing appropriate integration paths. Several conservative numerical schemes and approximate Riemann solvers have been generalised to non-conservative systems based on the theory by Dal Maso et al [22] : Lax-Friedrichs and Lax-Wendroff [24] , Roe's approximate Riemann solver [25] , HLL [26] and the Osher Riemann solver [27] . Parés [28] introduced the concept of path-conservative numerical schemes, as a generalisation of conservative schemes.…”

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

“…Moreover, the relaxation technique introduced here defines a general strategy suitable for both conservative and non-conservative systems, which could be of interest for the approximation of similar models such as the two-layer shallow flow model. Let us mention that at the time of revising this paper we became aware of a recent work of Castro et al [58] presenting an extension of Lax-Friedrichs scheme for general non-conservative systems that rigorously preserves positivity. We expect that our (first-order) method is less diffusive than this Lax-Friedrichs method, based on the comparative results in [58] between Lax-Friedrichs and Roe.…”

“…Let us mention that at the time of revising this paper we became aware of a recent work of Castro et al [58] presenting an extension of Lax-Friedrichs scheme for general non-conservative systems that rigorously preserves positivity. We expect that our (first-order) method is less diffusive than this Lax-Friedrichs method, based on the comparative results in [58] between Lax-Friedrichs and Roe.…”