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We propose and analyze a novel symmetric exponential wave integrator (sEWI)for the nonlinear Schr\"odinger equation (NLSE) with low regularity potentialand typical power-type nonlinearity of the form $ f(\rho) = \rho^\sigma $,where $ \rho:=|\psi|^2 $ is the density with $ \psi $ the wave function and $\sigma > 0 $ is the exponent of the nonlinearity. The sEWI is explicit andstable under a time step size restriction independent of the mesh size. Werigorously establish error estimates of the sEWI under various regularityassumptions on potential and nonlinearity. For "good" potential andnonlinearity ($H^2$-potential and $\sigma \geq 1$), we establish an optimalsecond-order error bound in $L^2$-norm. For low regularity potential andnonlinearity ($L^\infty$-potential and $\sigma > 0$), we obtain a first-order$L^2$-norm error bound accompanied with a uniform $H^2$-norm bound of thenumerical solution. Moreover, adopting a new technique of \textit{regularitycompensation oscillation} (RCO) to analyze error cancellation, for somenon-resonant time steps, the optimal second-order $L^2$-norm error bound isproved under a weaker assumption on the nonlinearity: $\sigma \geq 1/2$. Forall the cases, we also present corresponding fractional order error bounds in$H^1$-norm, which is the natural norm in terms of energy. Extensive numericalresults are reported to confirm our error estimates and to demonstrate thesuperiority of the sEWI, including much weaker regularity requirements onpotential and nonlinearity, and excellent long-time behavior withnear-conservation of mass and energy.

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