Global error estimates for exponential splitting

Global errors of principal exponential splitting formulae, i.e.,first-order splitting, Strang's splitting and parallel splitting,are investigated. It is found that the commutativity of theunderlying matrices in the exponents plays an important rolein the analysis. It is shown that the global error estimatesbehave as polynomials in the splitting steps when the stepsare small and decay exponen tially when the splitting stepsare chosen to be large. They are much more reliable than thosederived from the traditional local truncation error analysiswhen relatively large splitting steps are adopted. This corresponds,for example, to solving partial differential equations usingrelatively large Courant numbers. In this case, the global erroranalysis provides sharp bounds in contrast to the local errorestimates, which give inaccurate or even deceptive results.Numerical examples are given to demonstrate our results.