High order direct Arbitrary-Lagrangian-Eulerian (ALE) P N P M schemes with WENO Adaptive-Order reconstruction on unstructured meshes

Highlights: • High order $P_N{P}_M$ schemes on moving unstructured meshes. • $P_N{P}_M$ schemes applied to discontinuous solutions with $N>0$. • WENO Adaptive-Order reconstruction for reconstructed DG schemes. • WENO Adaptive-Order strategy as DG limiter. • Shock detector for DG limiting. -- Abstract: In this work we present a conservative WENO Adaptive Order (AO) reconstruction operator applied to an explicit one-step Arbitrary-Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) method. The spatial order of accuracy is improved by reconstructing higher order piecewise polynomials of degree $M>N$, starting from the underlying polynomial solution of degree N provided by the DG scheme. High order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor that arises from a one-step time integration procedure. As a result, space-time polynomials of order $M+1$ are obtained and used to perform the time evolution of the numerical solution adopting a fully explicit DG scheme. To maintain algorithm simplicity, the mesh motion is restricted to be carried out using straight lines, hence the old mesh configuration at time $t^n$ is connected with the new one at time $t^{n+1}$ via space-time segments, which result in space-time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our algorithm falls into the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where the governing PDE system is directly discretized relying on a space-time conservation formulation and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. A local rezoning strategy might be used in order to locally optimize the mesh quality and avoiding the generation of invalid elements with negative determinant. The proposed approach reduces to direct ALE finite volume schemes if $N=0$, while explicit direct ALE DG schemes are recovered in the case of $N=M$. In order to stabilize the DG solution, an a priori WENO based limiting technique is employed, that makes use of the numerical solution inside the element under consideration and its neighbor cells to find a less oscillatory polynomial approximation. By using a modal basis in a reference element, the evaluation of the oscillation indicators is very easily and efficiently carried out, hence allowing higher order modes to be properly limited, while leaving the zero-th order mode untouched for ensuring conservation. Numerical convergence rates for $2\le N,M\le 4$ are presented as well as a wide set of benchmark test problems for hydrodynamics on moving and fixed unstructured meshes.