FORCE Schemes on Moving Unstructured Meshes for Hyperbolic Systems

The aim of this paper is to propose a new simple and robust numerical flux of the centered type in the context of Arbitrary-Lagrangian–Eulerian (ALE) finite volume schemes. The work relies on the FORCE flux of Toro and Billet and is concerned with the solution of general hyperbolic systems of nonlinear equations involving both conservative and nonconservative terms as well as sources which might become stiff. The proposed approach is formulated in a general way using a path-conservative method and the Roe-type system matrix is computed numerically in order to provide a numerical flux function that can be applied to any given hyperbolic system. Furthermore, one great advantage of the FORCE flux is that no information about the eigenstructure of the system is needed, not even eigenvalues, but only information regarding the geometry of the control volumes are required, which are automatically available in the moving mesh framework. Our method is of the finite volume type, high order accurate in space, thanks to a WENO reconstruction operator, and even in time, due to a fully-discrete ADER one-step discretization. The algorithm applies to moving multidimensional unstructured meshes composed by triangles and tetrahedra. Both accuracy and robustness of the scheme are assessed on a series of test problems for the Euler equations of compressible gas dynamics, for the magnetohydrodynamics equations as well as for the Baer–Nunziato model of compressible multi-phase flows.