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Balanced Splitting and Rebalanced Splitting

Author(s)
Speth, Raymond L.; Green, William H.; MacNamara, Shevarl; Strang, Gilbert
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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.
Date issued
2013-11
URI
http://hdl.handle.net/1721.1/86001
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
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
ISSN
0036-1429
1095-7170

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