High order methods for the integration of the Bateman equations and other problems of the form of y′ = F(y,t)y

Abstract

This paper introduces two families of A-stable algorithms for the integration of y′=F(y,t)y: the extended predictor–corrector (EPC) and the exponential–linear (EL) methods. The structure of the algorithm families are described, and the method of derivation of the coefficients presented. The new algorithms are then tested on a simple deterministic problem and a Monte Carlo isotopic evolution problem.The EPC family is shown to be only second order for systems of ODEs. However, the EPC-RK45 algorithm had the highest accuracy on the Monte Carlo test, requiring at least a factor of 2 fewer function evaluations to achieve a given accuracy than a second order predictor–corrector method (center extrapolation / center midpoint method) with regards to Gd-157 concentration.Members of the EL family can be derived to at least fourth order. The EL3 and the EL4 algorithms presented are shown to be third and fourth order respectively on the systems of ODE test. In the Monte Carlo test, these methods did not overtake the accuracy of EPC methods before statistical uncertainty dominated the error.The statistical properties of the algorithms were also analyzed during the Monte Carlo problem. The new methods are shown to yield smaller standard deviations on final quantities as compared to the reference predictor–corrector method, by up to a factor of 1.4.