Statistical analysis of three fourth-order ordinary differential equation solvers

Abstract

We develop an autoregressive integrated moving average model (ARIMA) to study the statistical behavior of the numerical error generated from three fourth-order ODE Solvers: Milne's method, Adams-Bashforth method and a new method which randomly switches between Milne and Adams-Bashforth methods. With the actual error data based on three differential equations we desire to identify an ARIMA model to each data series. Results show that some of data series can be described by ARIMA models but others can not. Based on the mathematical form of the error data, other statistical models should be applied in the future. Finally we assess the multivariate normality of sample mean vectors which are generated by the switching method as an application of the multivariate central limit theorem.