Embedded exponential operator splitting methods for the time integration of nonlinear evolution equations

In this work, we introduce embedded exponential operator splitting methods for the adaptive time integration of nonlinear evolution equations. In the spirit of embedded Runge–Kutta methods, such pairs of related higher-order split-step methods provide estimates of the local error with moderate additional computational effort as substeps of the basic integrator are reused to obtain a local error estimator. ustrations, we construct a split-step pair of orders 4(3) involving real method coefficients, tailored for nonlinear Schrödinger equations, and two order 4(3) split-step pairs with complex method coefficients, appropriate for nonlinear parabolic problems. Our theoretical investigations and numerical examples show that the splitting methods retain their orders of convergence when applied to evolution problems with sufficiently regular solutions. Furthermore, we demonstrate the ability of the new algorithms to serve as a reliable basis for error control in the time integration of nonlinear evolution equations by applying them to the solution of two model problems, the two-dimensional cubic Schrödinger equation with focusing singularity and a three-dimensional reaction–diffusion equation. Moreover, we demonstrate the advantages of our real embedded 4(3) pair of splitting methods over a pair of unrelated schemes for the time-dependent Gross–Pitaevskii equation.