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Preference and Stability Regions for Semi-Implicit Composition Schemes

Author

Listed:
  • Petr Fedoseev

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

  • Artur Karimov

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

  • Vincent Legat

    (Institute of Mechanics, Materials and Civil Engineering (IMMC), Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium)

  • Denis Butusov

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

Abstract

A numerical stability region is a valuable tool for estimating the practical applicability of numerical methods and comparing them in terms of stability. However, only a little information can be obtained from the stability regions when their shape is highly irregular. Such irregularity is inherent to many recently developed semi-implicit and semi-explicit methods. In this paper, we introduce a new tool for analyzing numerical methods called preference regions. This allows us to compare various methods and choose the appropriate stepsize for their practical implementation, such as stability regions, but imposes stricter conditions on the methods, and therefore is more accurate. We present a thorough stability and preference region analysis for a new class of composition methods recently proposed by F. Casas and A. Escorihuela-Tomàs. We explicitly show how preference regions, plotted for an arbitrary numerical integration method, complement the conventional stability analysis and offer better insights into the practical applicability of the method.

Suggested Citation

  • Petr Fedoseev & Artur Karimov & Vincent Legat & Denis Butusov, 2022. "Preference and Stability Regions for Semi-Implicit Composition Schemes," Mathematics, MDPI, vol. 10(22), pages 1-13, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4327-:d:976981
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    References listed on IDEAS

    as
    1. R. Cavoretto & A. Rossi & M. S. Mukhametzhanov & Ya. D. Sergeyev, 2021. "On the search of the shape parameter in radial basis functions using univariate global optimization methods," Journal of Global Optimization, Springer, vol. 79(2), pages 305-327, February.
    2. Fernando Casas & Alejandro Escorihuela-Tomàs, 2020. "Composition Methods for Dynamical Systems Separable into Three Parts," Mathematics, MDPI, vol. 8(4), pages 1-18, April.
    3. Denis Butusov, 2021. "Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers," Mathematics, MDPI, vol. 9(9), pages 1-14, April.
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