OPTIMIZED ARTIFICIAL DISSIPATION TERMS FOR ENHANCED STABILITY LIMITS

Abstract

Finite difference approximations for the convection equation are developed, which exhibit enhanced stability limits for explicit Runge–Kutta integration. Stability limits are increased by adding artificial dissipation terms, which are optimized to yield greatest stable time steps. For the artificial dissipation terms, symmetric finite difference approximations of even-order derivatives are used with differencing stencils equal to the convective stencils. The spatial discretization inclusive of the added dissipation term is shown to be consistent with a first derivative. The formal order of accuracy in space is decreased by one order, while the order of time integration is not affected. As a result, the time step limits of originally stable Runge–Kutta integration is increased, for some combinations of spatial discretization and time integration by a factor of two. Algorithms, which are unstable without damping are stabilized. The dispersion properties of the algorithms are not influenced by the proposed damping terms. Spectral analysis of the algorithms show very low dissipation error for dimensionless wave numbers k Δ x < 0.5. Stability conditions based on von Neumann stability analysis are given for the proposed schemes for explicit Runge–Kutta time integration of orders up to ten.