Optimized integrating factor technique for Schrödinger-like equations

The integrating factor technique is widely used to solve numerically (in particular) the Schrödinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge condition of the potential. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We illustrate this approach with the nonlinear Schrödinger (NLS) and with the Schrödinger–Newton (SN) equations. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. This gain is crucial for long time simulations, as, with larger time steps, less computations are performed and the overall accumulation of round-off errors is reduced.