Randomized exponential integrators for modulated nonlinear Schrödinger equations
Provides a numerical method for solving PDEs with low-regularity coefficients, addressing a bottleneck in computational physics.
The paper develops a randomized exponential integrator for nonlinear Schrödinger equations with rough time-dependent dispersion, achieving convergence rates of order α+1/2 and handling more general modulations than prior methods.
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.