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Balanced splitting and rebalanced splitting. (English) Zbl 1284.65121

Summary: 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.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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