Efficient adaptive exponential time integrators for nonlinear Schrödinger equations with nonlocal potential

The performance of exponential-based numerical integrators for the time propagation of the equations associated with the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method for the approximation of the multi-particle Schrödinger equation in one space dimension is assessed. Among the most popular integrators such as Runge–Kutta methods, time-splitting, exponential integrators and Lawson methods, exponential Lawson multistep methods with one predictor–corrector step provide the best stability and accuracy at the least effort. This assessment is based on the observation that the evaluation of the nonlocal terms associated with the potential is the computationally most demanding part of such a calculation in our setting. In addition, the predictor step provides an estimator for the local time-stepping error, thus allowing for adaptive time-stepping which reflects the smoothness of the solution and enables to reliably control the accuracy of a computation in a robust way, without the need to guess an optimal stepsize a priori. One-dimensional model examples are studied to compare different time integrators and demonstrate the successful application of our adaptive methods.