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SUMMARYThis paper gives a comparative study of TVD-limiters for standard explicit Finite Volume schemes. In contrast to older studies, it includes also unsymmetrical limiter functions that depend on the local CFLnumber. We classify the limiters and show how to extend these families of limiters. We introduce a new member of the Superbee family, which is adapted to Roe's linear third-order scheme. Based on an idea by Serna and Marquina, new smooth limiters are introduced, which turn the van Leer and van Albada limiters into complete classes of limiters. The comparison of the limiters is done with some standard test cases. The results clarify the influence of the chosen limiter on the quality of the numerical results. Compared with ENO or WENO schemes, they also show the high resolution, which can be obtained by a CFL-number-dependent limiter when the grid is not highly refined.

In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The mathematical model is a simplified case of the seven-equation Baer-Nunziato model of compressible multi-phase flows. The resulting governing PDE system is a nonlinear system of hyperbolic conservation laws with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary inside the fluid phase. Due to the diffuse interface nature of the model, the volume fraction function can assume any value between zero and one in mixed cells that are occupied by both, fluid and solid.We also prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso, Le Floch and Murat [89], which justifies the use of a path-conservative approach for treating the non-conservative products.The governing partial differential equations of our new model are solved on simple uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) finite element method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion.The effectiveness of the proposed approach is tested on a set of different numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies.Key words: diffuse interface model, compressible flows over fixed and moving solids, immersed boundary method for compressible flows, arbitrary high-order discontinuous Galerkin schemes, a posteriori sub-cell finite volume limiter (MOOD), path-conservative schemes for hyperbolic PDE with non-conservative products,

Keywords: Double Mach reflection, high speed flow, high resolution scheme, numerical test case MSC 2010: 76J20, 76L05, 76M99This note discusses the initial and boundary conditions as well as the size of the computational domain for the double Mach reflection problem when set up as a test for the resolution of an Euler scheme for gas dynamics.

A classic problem in gas dynamics simulation is the occurrence of the carbuncle phenomenon, a breakdown of discrete shock profiles. We show that for high Froude number, this also occurs in shallow water simulations. Numerical evidence is given that commonly accepted cures developed for the numerics of gas dynamics should also work for shallow water flows.