A direct Arbitrary-Lagrangian–Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D

• Published in: Journal of computational physics Vol. 275; pp. 484 - 523
• Main Authors: Boscheri, Walter, Dumbser, Michael
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.10.2014
• Subjects: Algorithms; Arbitrary-Lagrangian–Eulerian (ALE) finite volume schemes; Baer–Nunziato model; Compressible multi-phase flows; Computation; Conservation laws and non-conservative hyperbolic PDE; Euler equations; Finite element method; High order of accuracy in space and time; Local rezoning; Magnetohydrodynamics; Mathematical analysis; Mathematical models; MHD equations; Partial differential equations; Stiff source terms; Three dimensional; WENO reconstruction on moving unstructured tetrahedral meshes; MHD equations; High order of accuracy in space and time; Stiff source terms; Local rezoning; Baer–Nunziato model; Arbitrary-Lagrangian–Eulerian (ALE) finite volume schemes; Euler equations; Conservation laws and non-conservative hyperbolic PDE; WENO reconstruction on moving unstructured tetrahedral meshes; Compressible multi-phase flows
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.06.059

In this paper we present a new family of high order accurate Arbitrary-Lagrangian–Eulerian (ALE) one-step ADER-WENO finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations with stiff source terms on moving tetrahedral meshes in three space dimensions. A WENO reconstruction technique is used to achieve high order of accuracy in space, while an element-local space–time Discontinuous Galerkin finite element predictor on moving curved meshes is used to obtain a high order accurate one-step time discretization. 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. Since our algorithm is cell-centered, the final mesh motion is computed by using a suitable node solver algorithm. A rezoning step as well as a flattener strategy are used in some of the test problems to avoid mesh tangling or excessive element deformations that may occur when the computation involves strong shocks or shear waves. The ALE algorithm presented in this article belongs to the so-called direct ALE methods because the final Lagrangian finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, with the rezoned geometry taken already into account during the computation of the fluxes. We apply our new high order unstructured ALE schemes to the 3D Euler equations of compressible gas dynamics, for which a set of classical numerical test problems has been solved and for which convergence rates up to sixth order of accuracy in space and time have been obtained. We furthermore consider the equations of classical ideal magnetohydrodynamics (MHD) as well as the non-conservative seven-equation Baer–Nunziato model of compressible multi-phase flows with stiff relaxation source terms. •Better than second order unstructured cell-centered Lagrangian finite volume schemes.•High order WENO reconstruction on moving unstructured tetrahedral meshes in 3D.•High order Lagrangian schemes for hyperbolic PDE with stiff source terms.•Path-conservative Lagrangian schemes for non-conservative hyperbolic PDE.•Applications to the Euler and MHD equations and to the Baer–Nunziato model in 3D.