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

doi:10.1007/s42967-023-00314-5

Abstract

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

CLC Number:

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.

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