3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method

Abstract

We describe a numerical scheme to solve 3D Arbitrary Lagrangian-Eulerian (ALE) hydrodynamics on an unstructured mesh using discontinuous finite element space and an explicit Runge-Kutta time discretization. This scheme combines the accuracy of a higher-order Godunov scheme with the unstructured mesh capabilities of finite elements that can be explicitly evolved in time. The spatial discretization uses trilinear isoparametric elements (tetrahedrons, pyramids, prisms and hexahedrons) in which the primitive variables (mass density, velocity and pressure) are piecewise trilinear. Upwinding is achieved by using Roe's characteristic decomposition of the inter-element boundary flux depending on the sign of characteristic wave speeds. The characteristics are evaluated at the Roe average, of variables on both sides of the inter-element boundary, for a general equation of state. An explicit second order Runge-Kutta time stepping is used for the time discretization. To capture shocks, we have generalized van Leer's 1D nonlinear minmod slope limiter to 3D using a quadratic programming scheme. For very strong shocks we find it necessary to supplement this with a Godunov stabilization where the trilinear representation of the variables is reduced to its constant average value. The resulting numerical scheme has been tested on a variety of problems relevant to ICF (inertial confinement fusion) target design and appears to be robust. It accurately captures shocks and contact discontinuities without unstable oscillations and has second-order accuracy in smooth regions. Object-oriented programming with the C++ programming language was used to implement our numerical scheme. The object-oriented design allows us to remove the complexities of an unstructured mesh from the basic physics modules and thereby enables efficient code development.