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Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

Author

Listed:
  • Aleksandra Tutueva

    (Department of Computer-Aided Design, Saint Petersburg Electrotechnical University “LETI”, 197376 Saint Petersburg, Russia)

  • Timur Karimov

    (Youth Research Institute, Saint Petersburg Electrotechnical University “LETI”, 197376 Saint Petersburg, Russia)

  • Denis Butusov

    (Youth Research Institute, Saint Petersburg Electrotechnical University “LETI”, 197376 Saint Petersburg, Russia)

Abstract

Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

Suggested Citation

  • Aleksandra Tutueva & Timur Karimov & Denis Butusov, 2020. "Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods," Mathematics, MDPI, vol. 8(5), pages 1-10, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:780-:d:357296
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    References listed on IDEAS

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    1. Nepomuceno, Erivelton G. & Rodrigues Junior, Heitor M. & Martins, Samir A.M. & Perc, Matjaž & Slavinec, Mitja, 2018. "Interval computing periodic orbits of maps using a piecewise approach," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 67-75.
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    1. Tutueva, Aleksandra V. & Nepomuceno, Erivelton G. & Karimov, Artur I. & Andreev, Valery S. & Butusov, Denis N., 2020. "Adaptive chaotic maps and their application to pseudo-random numbers generation," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).

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