Implicit second and third orders runge-kutta for handling discontinuities in delay differential equations

Abstract

Implicit Runge-Kutta (RK) methods have been developed and implemented in solving Delay Differential Equations (DDEs) systems which often encounter discontinuities. These discontinuities might occur after and even before the initial solution. The methods are chosen because they can be modified to handle discontinuities by means of mapping of past values and they are in fact the most well-organized way to handle the so-called stiff differential equations, which are differential equations usually characterized by a rapidly decaying solution. The advantage of implicit Runge-Kutta methods is in their superior stability compared to the explicit methods, more so when applied to stiff equations. Our objective is to develop a scheme for solving DDEs using implicit RK2 and RK3. Our numerical scheme is able to successfully handle discontinuities in the system and produces results with acceptable error. We compare the result from [1] which used explicit RK2 and RK4 with our findings. Our result is markedly better than [1] even in the lower order RK.