A robust and contact resolving Riemann solver on unstructured mesh, Part I, Euler method

Abstract

This article presents a new cell-centered numerical method for compressible flows on arbitrary unstructured meshes. A multi-dimensional Riemann solver based on the HLLC method (denoted by HLLC-2D solver) is established. The work is an extension from the cell-centered Lagrangian scheme of Maire et al. [27] to the Eulerian framework. Similarly to the work in [27], a two-dimensional contact velocity defined on a grid node is introduced, and the motivation is to keep an edge flux consistency with the node velocity connected to the edge intrinsically. The main new feature of the algorithm is to relax the condition that the contact pressures must be same in the traditional HLLC solver. The discontinuous fluxes are constructed across each wave sampling direction rather than only along the contact wave direction. The two-dimensional contact velocity of the grid node is determined via enforcing conservation of mass, momentum and total energy, and thus the new method satisfies these conservation properties at nodes rather than on grid edges. Other good properties of the HLLC-2d solver, such as the positivity and the contact preserving, are described, and the two-dimensional high-order extension is constructed employing MUSCL type reconstruction procedure. Numerical results based on both quadrilateral and triangular grids are presented to demonstrate the robustness and the accuracy of this new solver, which shows it has better performance than the existing HLLC method.