Operator Splitting

Abstract

Operator splitting is a numerical method of computing the solution to a differential equation. The splitting method separates the original equation into two parts over a time step, separately computes the solution to each part, and then combines the two separate solutions to form a solution to the original equation. A canonical example is splitting of diffusion terms and convection terms in a convection-diffusion partial differential equation. Related applications of splitting for reaction-diffusion partial differential equations in chemistry and in biology are emphasized here. The splitting idea generalizes in a natural way to equations with more than two operators. In all cases, the computational advantage is that it is faster to compute the solution of the split terms separately, than to compute the solution directly when they are treated together. However, this comes at the cost of an error introduced by the splitting, so strategies have been devised to control this error. This chapter introduces splitting methods and surveys recent developments in the area. An interesting perspective on absorbing boundary conditions in wave equations comes via Toeplitz-plus-Hankel splitting. One recent development, balanced splitting, deserves and receives special mention: it is a new splitting method that correctly captures steady state behavior.