Minimum Error Bounds for Local Truncation Errors to Iteratively Solve Scalar and Vector-Valued ODEs

Abstract

This paper aims at finding minimum error bounds on step size for a few second order explicit iterative techniques. Three standard most widely used Runge-Kutta type techniques and two recently developed iterative techniques are selected for analysis. Computation and examination of principal local truncation error and error bounds on step length reveal that minimum error bound is attained by Heun, Modified Improved Modified Euler (MIME) as derived in (Ma, 2010) and proposed iterative techniques and the same is confirmed by both scalar and vector-valued numerical problems given. Further, proposed iterative technique is found to have contained smallest local truncation error. Numerical calculations have been carried out using MATLAB version 8.1 (R2013a) in double precision arithmetic.