Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 1: 95 - 114

DOI: https://doi.org/10.1007/s42967-023-00314-5

Beyond Strang: a Practical Assessment of Some Second-Order 3-Splitting Methods

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  • 1 Department of Computer Science, University of Saskatchewan, 110 Science Place, Saskatoon S7N 5K9, SK, Canada;
    2 Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon S7N 5E6, SK, Canada;
    3 Department of Physics and Engineering Physics, University of Saskatchewan, 116 Science Place, Saskatoon S7N 5E2, SK, Canada

Received date: 2023-01-06

  Revised date: 2023-08-02

  Accepted date: 2023-09-02

  Online published: 2025-04-21

Supported by

The authors gratefully acknowledge funding from the Natural Sciences and Engineering Research Council of Canada under its Discovery Grant Program (RGPN 2020-04467 (RJS) and RGPN 2022-04482 (AS)) as well as from the US Air Force Office of Scientific Research FA9550-21-1-0031 (AS). A. Tavassoli acknowledges the support of Dr. Magdi Shoucri in developing the semi-Lagrangian code.

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Abstract

Operator-splitting is a popular divide-and-conquer strategy for solving differential equations. Typically, the right-hand side of the differential equation is split into a number of parts that are then integrated separately. Many methods are known that split the righthand side into two parts. This approach is limiting, however, and there are situations when 3-splitting is more natural and ultimately more advantageous. The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators. It is arguably the most popular method for 3-splitting because of its efficiency, ease of implementation, and intuitive nature. Other 3-splitting methods exist, but they are less well known, and analysis and evaluation of their performance in practice are scarce. We demonstrate the effectiveness of some alternative 3-splitting, second-order methods to Strang splitting on two problems: the reaction-diffusion Brusselator, which can be split into three parts that each have closed-form solutions, and the kinetic Vlasov-Poisson equations that are used in semi-Lagrangian plasma simulations. We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10%–20% over traditional Strang splitting. Our analysis for the practical assessment of the efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design.

Key words： Operator-splitting methods; Fractional-step methods; Brusselator; VlasovPoisson equations

Cite this article

Raymond J. Spiteri, Arash Tavassoli, Siqi Wei, Andrei Smolyakov . Beyond Strang: a Practical Assessment of Some Second-Order 3-Splitting Methods[J]. Communications on Applied Mathematics and Computation, 2025 , 7(1) : 95 -114 . DOI: 10.1007/s42967-023-00314-5

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