Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part III: The nonlinear case

The present work is concerned with the efficient time integration of nonlinear evolution equations by exponential operator splitting methods. Defect-based local error estimators serving as a reliable basis for adaptive stepsize control are constructed and analyzed. In the context of time-dependent nonlinear Schrödinger equations, asymptotical correctness of the local error estimators associated with the first-order Lie–Trotter and second-order Strang splitting methods is proven. Numerical examples confirm the theoretical results and illustrate the performance of adaptive stepsize control.