An explicit and symmetric exponential wave integrator for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity

Abstract

We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form $ |\psi|^{2\sigma}\psi $ with $ \psi $ being the wave function and $ \sigma > 0 $ being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For ``good" potential and nonlinearity ($H^2$-potential and $\sigma \geq 1$), we establish an optimal second-order error bound in the $L^2$-norm. For low regularity potential and nonlinearity ($L^\infty$-potential and $\sigma > 0$), we obtain a first-order $L^2$-norm error bound accompanied with a uniform $H^2$-norm bound of the numerical solution. Moreover, adopting a new technique of \textit{regularity compensation oscillation} (RCO) to analyze error cancellation, for some non-resonant time steps, the optimal second-order $L^2$-norm error bound is proved under a weaker assumption on the nonlinearity: $\sigma \geq 1/2$. For all the cases, we also present corresponding fractional order error bounds in the $H^1$-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.