Higher-Order Implicit Hybrid Runge-Kutta Schemes for Direct Numerical Solution of Third-Order Initial Value Problems

Abstract

Direct numerical solution of higher-order initial value problems remains an important problem in the numerical solution of ordinary differential equations particularly when conventional approaches require the reduction of higher-order equations to equivalent systems of first-order equations. This study develops two high-order implicit hybrid Runge-Kutta type schemes for the direct numerical solution of third-order initial value problems through the reformulation of linear multistep methods. Power series interpolation is employed to construct the two linear multistep method formulations, which are subsequently transformed into implicit hybrid Runge-Kutta type schemes with step lengths k = 3   and k = 4. The resulting methods are analyzed in terms of consistency, order, zero stability and convergence. The analysis establishes that the proposed schemes possess orders eight and nine for k = 3   and k = 4, respectively. The methods are applied to selected linear and nonlinear third-order initial value problems with known exact solutions, and their accuracy is assessed using absolute errors and comparative numerical experiments with existing methods. The numerical results demonstrate that both schemes provide accurate solutions, with the k = 4 generally yielding smaller absolute errors than the k = 3  formulation.