New efficient numerical methods for some systems of linear ordinary differential equations

Abstract

In mathematics there are several problems arise that can be described by differential equations with particular, highly complex structure. Most of the time, we cannot produce the exact (analytical) solution of these problems, therefore we have to approximate them numerically by using some approximating method. The main aim of this paper is to create numerical methods, based on operator splitting, that well approximate the exact solution of the original ODE systems while having low computational complexity. Starting from an example, based on the relationship between the Lie–Trotter (sequential) and Strang–Marchuk splitting methods, we examine the properties of processed integrator methods. Then we generalize these methods and introduce the new extended processed methods. By examining the consistency and stability of these methods, we establish the one order higher convergence. However, these methods have a higher computational complexity, which we aim to reduce by introducing economic extended processed methods. In this case we show the lower computational complexity and prove the second-order convergence. In the end, we test the analyzed methods in three models: a large-scale linear model, a piecewise-linear model of flutter and the heat conduction equation. Runtimes and errors are also compared.