A high-precision finite element method for shock-tube calculations

A two-pass explicit scheme is developed in order to exploit some of the capabilities of finite difference modeling (FDM) for finite element modeling (FEM), which offers the opportunity to account for any type of geometry in fluid flow modeling. Features of the first-order upwind and the Lax-Wendroff high precision explicit finite difference algorithms are reviewed. A flux limiter is developed for FEM to serve as an analog for the single limiter function which has been defined for the various FDMs. It is shown that an antidiffusive limiter must be introduced into the weighting function which normally multiplies the time-derivative term in the variational equation. The two-pass scheme which results is demonstrated to be the equivalent of FDMs with five-point support. However, the present scheme is valid only for one-dimensional calculations and linear shape functions for shock tube flow phenomena. Further work is required for its use with nonlinear hyperbolic systems.