Optimal stability polynomials for splitting methods, with application to the time-dependent Schrödinger equation Original Research Article

We determine optimal stability polynomials p(x) for splitting method solutions of differential equations, building on previous work by López-Marcos, Sanz-Serna and Skeel (1996). The methods have a variety of stage numbers and are up to eighth order. Knowledge of p(x) allows construction of the most stable splitting methods for given complexity. As an illustration, we construct symplectic corrector algorithms C−1KC, where the kernel K is an m-stage splitting method) which approximate the solution of linear Hamiltonian systems. The kernels K that realize the optimal stability polynomials are found for this case. We also discuss the construction of correctors C, and find them for two particularly promising kernels. Numerical calculations for a time-dependent Schrödinger equation problem confirm the methodsʹ usefulness.