Efficient Embedded Diagonal Implicit Runge-Kutta Method for Directly Solving Third Order ODEs

Abstract

This paper presents two important contributions to the field of numerical analysis for third-order ordinary differential equations (ODEs). First, a new class of direct implicit Runge-Kutta (RK) processes, called RKTDIO, is introduced as solutions to third-order ODEs. Secondly, it develops the ERKTDIO method, which is an embedded pairwise diagonal implicit RK method. The study begins by introducing the theory of relevant-colored trees and B-series as fundamental concepts. By utilizing the order constraints, two RKTDIO methods are derived: a fifth-order method with three stages and a sixth-order method with four stages. In addition, an embedded method called ERKTDIO6(5) is derived, which has orders six and five. The derivation of the embedded method includes strategies to ensure that the higher-order method achieves high accuracy while the lower-order method provides optimal error estimates. To evaluate the effectiveness of the proposed methods, variable step-size codes are developed and applied to a set of specific third-order problems. The numerical evaluation involves converting the problems into a system of first-order ODEs and comparing the results with existing methods in terms of accuracy and function evaluations. The numerical demonstrations emphasise the superior performance and efficiency of the new methods in solving third-order ODEs. The comparative analysis shows the accuracy achieved by the higher-order method and the improved error estimation of the lower-order method. The results validate the efficacy of the proposed approaches and their potential for practical applications in various domains.