Narrow Results

This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

This paper presents details of finite volume and finite element numerical models based on unstructured triangular meshes that are used to solve the two-dimensional nonlinear shallow water equations (SWEs). The finite volume scheme uses Roe's approximate Riemann solver to evaluate the convection terms. Second order accuracy is achieved by means of the MUSCL approach with MinMod and VanAlbada limiters. The finite element model utilizes the Lax–Wendroff two-step scheme, which is second-order in space and time. The models are validated and their relative performance compared for several benchmark problems, including a hydraulic jump, and flows in converging and converging–diverging channels.