Runge–Kutta Time Step Selection for Flow Problems

Sammanfattning: Optimality is studied for Runge-Kutta iteration for solving steady-state and time dependent flow problems. For the former type an algorithm for determining locally optimal time steps is developed, based on the fact that the squared norm of the residual produced by an m-stage scheme is a 2m-degree polynomial, the coefficients of which can be computed from scalar products of Krylov subspace vectors.Under certain conditions on the system matrix, the algorithm is guaranteed to work and its time steps will converge to a global optimum. Furthermore, it will outperform the use of any constant step size. The algorithm is modified to work even when those conditions are not satisfied.Experiments are carried out for a set of Euler and Navier-Stokes problems, both on a single and multiple grids. The algorithm can be extended with optimization over all RK coefficients or discrete parameters like the number of stages or multigrid levels. For that purpose, a simple discrete optimization algorithm is suggested.For some time-dependent problems in one dimension it is shown that if the difference operators and the time steps are properly selected, the local accuracy can be made one order higher than the formal order of the difference operators suggests. This idea cannot be fully generalized, but it will work for scalar problems in 2D if it is combined with an alternating flow technique. Finally an error filter is developed that allows standard step size control algorithms for ordinary differential equations to be efficiently applied to partial differential equations involving shocks.