Strong Stability Preserving Explicit Runge-Kutta Methods of Maximal Effective Order

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construc- tion of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods—like classical order five methods— require the use of nonpositive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge-Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.