Embedded Split-Step Formulae for the Time Integration of Nonlinear Evolution Equations

We introduce pairs of related split-step time integration formulae for the numeri- cal solution of evolution equations. In the spirit of embedded Runge-Kutta pairs, these yield estimates of the local error with moderate additional computational effort as substeps of the basic splitting method are reused to provide a higher-order splitting formula used for error estimation. As an illustration, we derive an order 4(3) pair with real method coefficients and two complex examples of such formulae. We demonstrate the ability of these methods to serve as a reliable basis for error control in the integration of nonlinear evolution equations by applying them to the solution of the cubic Schrodinger equation and a nonlinear parabolic problem, respectively.