Exponential Runge–Kutta methods for the Schrödinger equation

We consider exponential Runge–Kutta methods of collocation type, and use them to solve linear and semi-linear Schrödinger Cauchy problems on the d-dimensional torus. We prove that in both cases (linear and non-linear) and with suitable assumptions, s-stage methods are of order s and we give sufficient conditions to achieve orders s+1 and s+2. We show and explain the effects of resonant time steps that occur when solving linear Schrödinger problems on a finite time interval with such methods. This work is inspired by [M. Hochbruck, A. Ostermann, Exponential Runge–Kutta methods for parabolic problems, Appl. Numer. Math. 53 (2–4) (2005) 323-339], where exponential Runge–Kutta methods of collocation type are applied to parabolic Cauchy problems. We compare our results with those obtained for parabolic problems and provide numerical experiments for illustration.