A survey of pseudo Runge-Kutta methods

This survey collects the theoretical results in the area of pseudo RungeKutta methods (PRK ) for ordinary differential equations and it is a vehicle for a current bibliography from 1966 to 2002. PRK methods require fewer functional evaluations than Runge-Kutta methods of the same order. Byrne and Lambert (1966-1967 ) was the first who considered PRK methods in significative forms. Afterwards Costabile (1968-1975 ) introduced PRK methods of II and III species as an alternative to the first ones. The latter methods are also autostarting and reduce the cost by 50 percent compared with the similar Runge-Kutta ones. Nakashima (1982-1999 ) improved PRK methods of II species and introduced the implicit methods. Jackiewicz, Tracogna, Bartoszewki, Zennaro, Wanner, Hairer (19912000 ) introduced the modern theory of order, also with variable step-size and embedded and continuous formulas. Finally Bartoszewki and Jackiewicz (2000 ) introduced a PRK code for nonstiff differential systems. PRK methods for special second order differential equations are also studied. 1 – Introduction For the numerical solution of the initial value problem (1) { y′(x) = f(x, y(x)) y(x0) = y0 where y : [x0, b] −→ IR , f : [x0, b] × IR −→ IR , y0 ∈ IR and f satisfies all hypotesis for existence and unicity of solution, the explicit