Derivation of three-derivative Runge-Kutta methods

Abstract

We introduce an algorithm for a numerical integration of ordinary differential equations in the form of yź = f(y). We extend the two-derivative Runge-Kutta methods (Chan and Tsai, Numer. Algor. 53, 171---194, 2010) to three-derivative Runge-Kutta methods by including the third derivative yźźź=ăź(y)=fźź(y)(f(y),f(y))+fź(y)fź(y)f(y)$y^{\prime \prime \prime }=\hat {g}(y)=f^{\prime \prime }(y)(f(y), f(y))+f^{\prime }(y)f^{\prime }(y)f(y)$. We present an approach based on the algebraic theory of Butcher (Math. Comp. 26, 79---106, 1972) and the źź$\mathcal {B}-$ series theory of Hairer and Wanner (Computing 13, 1---15 (1974)) combined with the methodology of Chan and Chan (Computing 77(3), 237---252, 2006). In this study, special explicit three-derivative Runge-Kutta methods that possess one evaluation of first derivative, one evaluation of second derivative, and many evaluations of third derivative per step are introduced. Methods with stages up to six and of order up to ten are presented. The numerical calculations have been performed on some standard problems and comparisons made with the accessible methods in the literature.