A comparison of two-step time integration schemes for the finite element advection equation

Numerical studies of two explicit, two step time integration techniques for the one dimensional, constant velocity finite element advection equation have been conducted for both square hill and cosine hill density distributions. One of these integration techniques, the Godunov scheme, is first order accurate in time while the other, the Lax-Wendroff scheme, is second order accurate in time. The results show that, overall, the "best" numerical solutions are obtained by combining a central weighted first step Lax-Wendroff time integration with parabolic spatial discretization either in its full or condensed M matrix form. Both the standard and central weighted first step Godunov time integrations are found to be numerically diffusive. This diffusivity tends to override whatever spatial discretization is used. However, the positivity property possessed by the Godunov schemes can be valuable for many applications.