High order accurate direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on moving curvilinear unstructured meshes

Abstract

In this article we present a new high order accurate fully discrete one-step Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving unstructured curvilinear meshes in two and three space dimensions. The WENO reconstruction technique that is used to achieve high order of accuracy in space is performed on curved isoparametric triangular and tetrahedral elements, which are not necessarily defined by straight boundaries. High order of accuracy in time is obtained via an element-local space-time Galerkin finite element predictor on moving curved meshes already developed in [Boscheri W, Dumbser M. A direct arbitrary-lagrangian-eulerian ader-weno finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3d. Journal of Computational Physics 2014;275(0):484–523.]. Our algorithm belongs to the category of cell-centered schemes, therefore a nodal solver is used to compute the velocity at each vertex of the computational grid, as well as at each additional degree of freedom that is needed to approximate the curvilinear geometry. To avoid mesh tangling or extremely distorted elements, we propose to use a modified version of the rezoning algorithm presented in [Galera S, Maire P, Breil J. A two-dimensional unstructured cell-centered multi-material ale scheme using vof interface reconstruction. Journal of Computational Physics 2010;229:5755-5787.], which can deal with curvilinear elements in multiple space dimensions. The rezoned geometry is then taken into account directly during the computation of the fluxes, thus the resulting finite volume scheme is a direct ALE method based on a space-time conservation formulation of the governing PDE system. The space-time control volume is defined for each element at each time step adopting an isoparametric approach, i.e. relying on a set of space-time basis functions which are as accurate as the desired order of the scheme. In this way the numerical solution and the geometry configuration of each element are approximated with the same accuracy in space and time. The resulting scheme is thus high order accurate and fully-discrete in one single step, which is typical for the ADER approach. We apply our new algorithm to the Euler equations of compressible gas dynamics in two and three space dimensions, considering a set of classical numerical test problems on moving meshes. Furthermore numerical convergence studies show the high order of accuracy of the proposed method up to fifth order in space and time.