Balanced Splitting and Rebalanced Splitting

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.