Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

Abstract

We establish uniform error estimates of finite difference methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter $\varepsilon$ ($\varepsilon\in(0,1]$). When $\varepsilon\to0^+$, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., $0<\varepsilon\ll1$, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with $O(\varepsilon^2)$-wavelength at $O(\varepsilon^4)$ and $O(\varepsilon^2)$ amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in $\varepsilon$ of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in $\varepsilon$, at the order of $O(h^2+\tau)$ and $O(h^2+\tau^{2/3})$ with time step $\tau$ and mesh size $h$ for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the $l^2$-norm and discrete semi-$H^1$ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine $\varepsilon$-dependent error estimates of NLSW, $\varepsilon$-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-$H^1$ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.