Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers

Abstract In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. in [13] to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space–time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allow for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. The space–time flux integral computation is carried out at the boundaries of each triangular space–time control volume using the Simpson quadrature rule in space and Gauss–Legendre quadrature in time. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. Since our one-step ALE finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, the remapping stage is not needed, making our algorithm a so-called direct ALE method. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach. Furthermore, the new high order Lagrangian schemes based on genuinely multidimensional Riemann solvers have been carefully compared with high order Lagrangian finite volume schemes based on conventional one-dimensional Riemann solvers. It has been clearly shown that due to the less restrictive CFL condition the new schemes based on multidimensional HLL and HLLC Riemann solvers are computationally more efficient than the ones based on a conventional one-dimensional Riemann solver technique.

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