3D UNSTRUCTURED MESH ALE HYDRODYNAMICS WITH THE UPWIND DISCONTINUOUS FINITE-ELEMENT METHOD

We describe a numerical scheme to solve 3D Arbitrary Lagrangian-Euleri an (ALE) hydrodynamics on an unstructured mesh using discontinuous fin ite element space and an explicit Runge-Kutta time discretization. Thi s scheme combines the accuracy of a higher-order Godunov scheme with t he unstructured mesh capabilities of finite elements that can be expli citly evolved in time. The spatial discretization uses trilinear isopa rametric elements (tetrahedrons, pyramids, prisms and hexahedrons) in which the primitive variables (mass density, velocity and pressure) ar e piecewise trilinear. Upwinding is achieved by using Roe's characteri stic 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 b oundary, for a general equation of state. An explicit second order Run ge-Kutta time stepping is used for the time discretization. To capture shocks, we have generalized van Leer's 1D nonlinear minmod slope limi ter to 3D using a quadratic progamming scheme. For very strong shocks we find it necessary to supplement this with a Godunov stabilization w here the trilinear representation of the variables is reduced to its c onstant 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 sec ond-order accuracy in smooth regions. Object-oriented programming with the C++ programming language was used to implement our numerical sche me. The object-oriented design allows us to remove the complexities of an unstructured mesh from the basic physics modules and thereby enabl es efficient code development. (C) 1998 Elsevier Science S.A.