On Error Estimation In General Linear Methods: Runge Kutta (Rk) And Almost Runge-Kutta (Ark) Methods

Abstract— General linear methods (GLM) apply to a large family of numerical methods for ordinary differential equations, with RungeKutta (RK) and Almost Runge-Kutta (ARK) methods as two out of a number of special cases. In this paper, we have investigated the efficacy of Richardson extrapolation (RE) technique as a means of obtaining viable and acceptable estimates of the local truncation errors (LTE) when explicit RK and ARK methods are used to solve non-stiff initial value problems (IVP). Results from numerical experiments conducted confirmed the popular idea that RE technique works for RK methods. However, experimental results for ARK methods; the first of its kind, show that RE does not work for explicit ARK methods. Keywords—estimation,error, general linear methods, RungeKutta, Almost Runge-Kutta, Richardson extrapolation, Viable, Acceptable