An Adaptive, Highly Accurate and Efficient, Parker-Sochacki Algorithm for Numerical Solutions to Initial Value Ordinary Differential Equation Systems

Abstract

The Parker-Sochacki Method (PSM) allows the numerical approximation of solutions to a polynomial initial value ordinary differential equation or system (IVODE) using an algebraic power series method. PSM is equivalent to a modified Picard iteration and provides an efficient, recursive computation of the coefficients of the Taylor polynomial at each step. To date, PSM has largely concentrated on fixed step methods. We develop and test an adaptive stepping scheme that, for many IVODEs, enhances the accuracy and efficiency of PSM. PSM Adaptive (PSMA) is compared to its fixed step counterpart and to standard Runge-Kutta (RK) foundation algorithms using three example IVODEs. In comparison, PSMA is shown to be competitive, often outperforming these methods in terms of accuracy, number of steps, and execution time. A library of functions is also presented that allows access to PSM techniques for many non-polynomial IVODEs without having to first rewrite these in the necessary polynomial form, making PSM a more practical tool.

A New Analytic Approximation of Luminosity Distance in Cosmology Using the Parker–Sochacki Method

“…The concept of polynomial projection may be better explained by referring to a simple example (taken from Ref. [43]). So consider the following first order nonlinear IVP…”