Abstract

In this work we develop a new class of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space-time finite element predictor, which performs a one-step time discretization. Specifically, we adopt either the continuous Galerkin (CG) predictor, which does not allow discontinuities in time and is suitable for smooth time evolutions, or the discontinuous Galerkin (DG) predictor which can handle stiff source terms that might produce jumps in the local space-time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space-time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The node solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration is connected with the new one at the future time level with straight edges to represent the local mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space-time divergence operator and is then integrated on a moving space-time control volume. We adopt a linear parametrization of the space-time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. In our algorithm either a simple and robust Rusanov-type numerical flux or a more sophisticated and less dissipative Osher-type numerical flux is employed. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical and relativistic magneto-hydrodynamics (MHD) equations and the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Furthermore we focus on the following issues to improve the algorithm efficiency: the time evolution, the numerical flux computation across element boundaries and the high order WENO reconstruction procedure. First, a time-accurate local time stepping (LTS) algorithm for unstructured triangular meshes is derived and presented, where each element can run at its own optimal time step, given by a local CFL stability condition. Then, we propose a new and efficient quadrature-free formulation for the flux computation, in which the space-time boundaries of each element are split into simplex sub-elements. This leads to space-time normal vectors as well as Jacobian matrices that are constant within each sub-element, hence allowing the flux integrals to be evaluated on the space-time reference control volume once and for all analytically during a preprocessing step. Finally, we consider the very new a posteriori MOOD paradigm, recently proposed for the Eulerian framework, to overcome the expensive WENO approach on moving meshes. The MOOD technique requires the use of only one central reconstruction stencil because the limiting procedure is carried out a posteriori instead of a priori, as done in the WENO formulation.