Error analysis of a class of semi-discrete schemes for solving the Gross–Pitaevskii equation at low regularity

Abstract

We analyze a class of time discretizations for solving the nonlinear Schrödinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain Ω⊂Rd, d≤3. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space Hr, r≥0, beyond the more typical L2 or Hσ(σ>d2) -error analysis. Numerical experiments illustrate our results.