Higher-order operator splitting methods for deterministic parabolic equations

Abstract

The Sheng-Suzuki theorem states that all exponential operator splitting methods of order greater than 2 must contain negative time integration. There have been claims in the literature that higher-order splitting methods for deterministic parabolic equations are unstable due to this fact. We show stability for a class of higher-order splitting methods for integrating deterministic parabolic equations. We note that problems with backwards time integration will still exist for stochastic integration methods for which information is lost and backward timesteps become ill-defined. Therefore, completely positive splitting methods, such as those developed by Chin, still have an important place. We present numerical results from first-, second-, third- and fourth-order methods showing that the error becomes increasingly small as the order increases.