Functional Continuous Runge-Kutta Methods with Reuse

TL;DR

This paper introduces new explicit functional continuous Runge-Kutta methods for retarded functional differential equations that reuse the last stage to reduce computational cost while maintaining high order accuracy.

Contribution

The paper develops novel Runge-Kutta methods that reuse the last stage, achieving higher efficiency for solving retarded functional differential equations.

Findings

Methods of orders three, four, and five with reduced computations are constructed.

Numerical tests confirm the convergence order and lower computational cost.

New methods outperform existing ones in efficiency for test problems.

Abstract

In the paper explicit functional continuous Runge-Kutta and Runge-Kutta-Nystr\"om methods for retarded functional differential equations are considered. New methods for first order equations as well as for second order equations of the special form are constructed with the reuse of the last stage of the step. The order conditions for Runge-Kutta-Nystr\"om methods are derived. Methods of orders three, four and five which require less computations than the known methods are presented. Numerical solution of the test problems confirm the convergence order of the new methods and their lower computational cost is performed.