Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part II: Comparisons of local error estimation and step-selection strategies for nonlinear Schrödinger and wave equations

We compare the practical performance of adaptive splitting methods for the solution of nonlinear Schrödinger equations. Different methods for local error estimation are assessed with respect to their accuracy and efficiency in conjunction with promising strategies for step-size adaptation. The numerical comparisons comprise the cubic nonlinear Schrödinger equation with a blow-up solution, systems of coupled nonlinear Schrödinger equations, a rotational and a Gross-Pitaevskii equation under an oscillatory potential inducing wave chaos, and a quantum control model with a time-dependent potential. Finally, for nonlinear wave equations we demonstrate the enhanced computational stability ensuing from adaptive step selection strategies close to the border mandated by the CFL condition.