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Splitting Methods in Communication, Imaging, Science, and Engineering pp 95–114Cite as

Operator Splitting

Part of the Scientific Computation book series (SCIENTCOMP)

Abstract

Operator splitting is a numerical method of computing the solution to a differential equation. The splitting method separates the original equation into two parts over a time step, separately computes the solution to each part, and then combines the two separate solutions to form a solution to the original equation. A canonical example is splitting of diffusion terms and convection terms in a convection-diffusion partial differential equation. Related applications of splitting for reaction-diffusion partial differential equations in chemistry and in biology are emphasized here. The splitting idea generalizes in a natural way to equations with more than two operators. In all cases, the computational advantage is that it is faster to compute the solution of the split terms separately, than to compute the solution directly when they are treated together. However, this comes at the cost of an error introduced by the splitting, so strategies have been devised to control this error. This chapter introduces splitting methods and surveys recent developments in the area. An interesting perspective on absorbing boundary conditions in wave equations comes via Toeplitz-plus-Hankel splitting. One recent development, balanced splitting, deserves and receives special mention: it is a new splitting method that correctly captures steady state behavior.

Keywords

  • Toeplitz Matrix
  • Operator Splitting
  • Absorb Boundary Condition
  • Finite Difference Approximation
  • Hankel Matrix

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 3.1

Notes

  1. 1.

    An exercise in Golub and Van Loan [22] shows that \([\boldsymbol{A},\boldsymbol{B}] =\boldsymbol{ 0}\) if and only if \(e^{h(\boldsymbol{A}+\boldsymbol{B})} = e^{h\boldsymbol{A}}e^{h\boldsymbol{B}}\) for all h.

  2. 2.

    See, for example, Rauch’s notes on Turing instability [48].

  3. 3.

    Here we assume \(\boldsymbol{A}\) and \(\boldsymbol{B}\) are invertible. The non-invertible case is treated by the variation-of-parameters formula [58].

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Author information

Authors and Affiliations

  1. School of Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia

    Shev MacNamara

  2. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Gilbert Strang

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  1. Shev MacNamara
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  2. Gilbert Strang
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Correspondence to Shev MacNamara .

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Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, California, USA

    Stanley J. Osher

  3. Department of Mathematics, UCLA, Los Angeles, California, USA

    Wotao Yin

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MacNamara, S., Strang, G. (2016). Operator Splitting. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_3

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