Central Difference TVD Schemes for Time Dependent and Steady State Problems

Abstract

We use central differences to solve the time dependent Euler equations. The schemes are all advanced using a Runge-Kutta formula in time. Near shocks a second difference is added as an artificial viscosity. This reduces the scheme to a first-order upwind scheme at shocks. The switch that is used guarantees that the scheme is TVD. For steady state problems it is usually advantageous to relax this condition. Then small oscillations do not activate the switches and the convergence to a steady state is improved. To sharpen the shocks different coefficients are needed for different equations, so a matrix-valued dissipation is introduced and compared with the scalar viscosity. The connection between this artificial viscosity and flux limiters is shown. Any flux limiter can be used as the basis of a shock detector for an artificial viscosity. We compare the use of the van Leer, van Albada, minmod, superbee, and the "average" flux limiters for this central difference scheme. For time dependent problems we need to use a small enough time step so that the CFL is less than one even though the scheme is linearly stable for larger time steps. Using a TVB Runge-Kutta scheme yields minor improvements in the accuracy.