Marvin Knöller

Marvin Knöller, Alexander Ostermann, Katharina Schratz

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schrödinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$ in order to be second-order convergent in $H^r$, i.e., it requires the boundedness of four additional derivatives of the solution. We present a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space. The latter can be efficiently evaluated by fast Fourier methods. For second-order convergence, the new integrator requires two additional derivatives of the solution in one space dimension, and three derivatives in higher space dimensions. Numerical examples illustrating our convergence results are included. These examples demonstrate the clear advantage of the Fourier integrator over standard Strang splitting for initial data with low regularity.

Martina Hofmanová, Marvin Knöller, Katharina Schratz

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{α,2}$ for some $α\in (0,1)$. Due to the loss of smoothness in the problem classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $α+1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.