Strong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order
KAUST DepartmentNumerical Mathematics Group
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
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AbstractWe 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 construction 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.
CitationStrong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order 2013, 51 (4):2149 SIAM Journal on Numerical Analysis