Balanced Splitting and Rebalanced Splitting
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Speth-2013-BALANCED SPLITTING A.pdf
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Author(s)
Speth, Raymond L.
Green, William H.
MacNamara, Shevarl
Strang, Gilbert
Date Issued
November 2013
Journal
SIAM Journal on Numerical Analysis
Publisher
Society for Industrial and Applied Mathematics
Citation
Speth, Raymond L., William H. Green, Shev MacNamara, and Gilbert Strang. “Balanced Splitting and Rebalanced Splitting.” SIAM Journal on Numerical Analysis 51, no. 6 (January 2013): 3084–3105. © 2013, Society for Industrial and Applied Mathematics
Version
Final published version
Abstract
Many systems of equations fit naturally in the form $du/dt = A(u) + B(u)$. We may separate convection from diffusion, $x$-derivatives from $y$-derivatives, and (especially) linear from nonlinear. We alternate between integrating operators for $dv/dt=A(v)$ and $dw/dt=B(w)$. Noncommutativity (in the simplest case, of $e^{Ah}$ and $e^{Bh}$) introduces a splitting error which persists even in the steady state. Second-order accuracy can be obtained by placing the step for $B$ between two half-steps of $A$. This splitting method is popular, and we suggest a possible improvement, especially for problems that converge to a steady state. Our idea is to adjust the splitting at each timestep to $[A(u) + c_n] + [B(u)-c_n]$. We introduce two methods, balanced splitting and rebalanced splitting, for choosing the constant $c_n$. The execution of these methods is straightforward, but the stability analysis becomes more difficult than for $c_n=0$. Experiments with the proposed rebalanced splitting method indicate that it is much more accurate than conventional splitting methods as systems approach steady state. This should be useful in large-scale simulations (e.g., reacting flows). Further exploration may suggest other choices for $c_n$ which work well for different problems.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Massachusetts Institute of Technology. Department of Chemical Engineering
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/120878641
