Parallel start for explicit parallel two-step peer methods

Abstract

Explicit parallel two-step peer methods use s stages with essentially identical properties. They are quite efficient in solving standard nonstiff initial value problems and may obtain a parallel speed-up near s on s processors for expensive problems. The two-step structure requires s − 1 initial approximations which have been computed by one-step methods in earlier versions. We now present a self-contained starting procedure using parallel Euler steps in the initial interval. Low order error terms introduced by this step are eliminated by special coefficient sets increasing the order to s after s − 2 time steps. An estimate for the initial stepsize is discussed, as well. Parallel OpenMP experiments with realistic problems demonstrate the efficiency compared to standard codes.