Balanced Splitting and Rebalanced Splitting

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 eAh and eBh) 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) + cₙ] + [B(u)-cₙ]. We introduce two methods, balanced splitting and rebalanced splitting, for choosing the constant cₙ. The execution of these methods is straightforward, but the stability analysis becomes more difficult than for cₙ=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 cₙ which work well for different problems.