High Phase-Lag-Order Runge--Kutta and Nyström Pairs

Abstract

We exploit the freedom in the selection of the free parameters of one family of eighth-algebraic-order Runge--Kutta (RK) pairs and of three families of fourth-, sixth-, and eighth-order RK Nyström (RKN) pairs with the purpose of obtaining specific pairs of the highest possible phase-lag order, which are also characterized by minimized principal truncation error coefficients. We present a method for the analytic derivation of the dissipation-order conditions for RK methods and the phase-lag- and dissipation-order conditions for Nyström methods. The RK pairs we study here are based on a one-parameter generalization of some older families of pairs. An algorithm and specific optimized 8(6) Nyström pairs are also provided. For a class of initial value problems, whose solution is known to be described by free oscillations or free oscillations of low frequency with forced oscillations of high frequency superimposed, over long integration intervals, these new pairs seem to offer some advantages with respect to some older pairs. The latter are of the same algebraic orders as the new ones but are characterized by the minimal phase-lag order according to their algebraic order and number of stages.