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Abstract
Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems.
Highlights
In recent years, tools for computer simulation have become increasingly important in various fields of science
To study the continuous dynamical systems described by ordinary differential equations (ODE), numerical integration methods are traditionally used
It is of interest to develop semi-implicit multistep methods based on numerical extrapolation techniques. We suggest closing this gap by introducing a novel multistep ODE solver combining the advantages of the extrapolation technique and the semi-implicit symmetric basic method
Summary
Tools for computer simulation have become increasingly important in various fields of science. The well-known problem of such integration is a decrease in numerical stability with an increase in the accuracy order of the chosen scheme This property restricts the application of high-order multistep methods for solving. The application of implicit methods involves computationally expensive approximation of the numerical solution at the point using Newton’s method or some other iterative technique This fact eliminates one of the main advantages of multistep algorithms—their computational simplicity and high efficiency. Efficient techniques for non-Hamiltonian systems [11] simulation were developed based on Aitken–Neville extrapolation and the semi-implicit symmetric basic method [12]. We suggest closing this gap by introducing a novel multistep ODE solver combining the advantages of the extrapolation technique and the semi-implicit symmetric basic method.
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