A Theoretical Investigation of Local Truncation Error and Convergence of an Accelerated Third-Order Runge–Kutta Method for Non-Autonomous Ordinary Differential Equations

Abstract

In this article, we theoretically examined well-organized third order numerical technique for IVP of ODE’s including partial derivative which has enhanced its competency regarding truncation error. Accelerated numerical method for local truncation error and convergence is theoretically investigated to assess how accurate and reliable proposed method is. Expansion of Taylor’s series is done that provides right pattern to expand and evaluate function evaluation. With the help of Taylor’s series, Local truncation error is an explicitly derived to clarify the order of accuracy. Linear standard test is discussed for calculating stability. Stability is investigated for knowing behavior of method. Stability region is drawn by MATLAB2023a software. Stability region is visualized to check numerical method possess a bounded solution when applied frequently and repeatedly. Consistency is proved which tells us error goes to zero as much as we decrease step size which guarantees the desired result. Convergence criteria theoretically discussed here by proving consistency and stability. Theoretically findings showing that the numerical method is stable, attains high accuracy and gives reliable performance. Therefore, method is efficient and applicable for extensive class of IVP arising in the area of ODE’s.