Low regularity exponential-type integrators for semilinear Schrödinger equations

— We introduce low regularity exponential-type integrators for nonlinear Schrödinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order convergence in H for solutions in H (r > d/2) of the derived schemes. This allows us lower regularity assumptions on the data than for instance required for classical splitting or exponential integration schemes. For one dimensional quadratic Schrödinger equations we can even prove first-order convergence without any loss of regularity. Numerical experiments underline the favorable error behavior of the newly introduced exponential-type integrators for low regularity solutions compared to classical splitting and exponential integration schemes. Semilinear Schrödinger equations, in particular those of type (1) i∂tu = −∆u+ μ|u|u, p ∈ N with μ ∈ R are nowadays extensively studied numerically. In this context, splitting methods (where the right-hand side is split into the kinetic and nonlinear part, respectively) as well as exponential integrators (including Lawson-type Runge–Kutta methods) contribute particularly attractive classes of integration schemes. For an extensive overview on splitting and exponential integration methods we refer to [15, 16, 18, 26], and for their rigorous convergence analysis in the context of semilinear Schrödinger equations we refer to [3, 4, 6, 8, 9, 11, 12, 25, 31] and the references therein. However, within the construction of all these numerical methods the stiff part (i.e., the terms involving the differential operator ∆) is approximated in one way or another which generally requires the boundedness of two additional spatial derivatives of the exact solution. In particular, convergence of a certain order only holds under sufficient additional regularity assumptions on the solution. In the following we will illustrate the local error behavior of classical splitting and exponential integration methods and the thereby introduced smoothness requirements.