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 the underlying matrices in the exponents plays an important role in the analysis. It is shown that the global error estimates behave as polynomials in the splitting steps when the steps are small and decay exponentially when the splitting steps are chosen to be large. They are much more reliable than those derived from the traditional local truncation error analysis when relatively large splitting steps are adopted. This corresponds, for example, to solving partial differential equations using relatively large Courant numbers. In this case, the global error analysis provides sharp bounds in contrast to the local error estimates, which give inaccurate or even deceptive results. Numerical examples are given to demonstrate our results.