Trigonometrically fitted three-derivative Runge–Kutta methods for solving oscillatory initial value problems

Abstract

Trigonometrically fitted three-derivative Runge–Kutta (TFTHDRK) methods for solving numerically oscillatory initial value problems are proposed and developed. TFTHDRK methods improve three-derivative Runge–Kutta (THDRK) methods [Numer. Algor. 74: 247–265, 2017] and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of {exp(iwt),exp(−iwt)} or equivalently the set {cos (wt), sin (wt)}, where w approximate the main frequency of the problem. The order conditions are deduced by the theory of rooted trees and B-series and two new explicit special TFTHDRK methods with order five and seven, respectively, are constructed. Linear stability of TFTHDRK methods is examined. Numerical results show the superiority of the new methods over other methods from the scientific literature.