An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high-order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space-time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element-local space-time Galerkin finite element predictor on moving curved meshes to obtain a high-order accurate one-step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tnis connected to the new one at tn + 1by straight edges, hence providing unstructured space-time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature-free integration, in which the space-time boundaries of the control volumes are split into simplex sub-elements that yield constant space-time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub-elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high-order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.