Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations
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We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schrödinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in treating the zeroth mode exactly within the discretization. For long-time error estimates, we rigorously establish the long-time error bounds of different low-regularity integrators for the nonlinear Schrödinger equation (NLSE) with small initial data characterized by a dimensionless parameter ε∈ (0, 1]. We begin with the low-regularity integrator for the quadratic NLSE in which the integral is computed exactly and the improved uniform first-order convergence in H^r is proven at O(ετ) for solutions in H^r with r > 1/2 up to the time T_ε = T/ε with fixed T > 0. Then, the improved uniform long-time error bound is extended to a symmetric second-order low-regularity integrator in the long-time regime. For the cubic NLSE, we design new non-resonant first-order and symmetric second-order low-regularity integrators which treat the zeroth mode exactly and rigorously carry out the error analysis up to the time T_ε = T/ε ^2. With the help of the regularity compensation oscillation (RCO) technique, the improved uniform error bounds are established for the new non-resonant low-regularity schemes, which further reduce the long-time error by a factor of ε^2 compared with classical low-regularity integrators for the cubic NLSE. Numerical examples are presented to validate the error estimates and compare with the classical time-splitting methods in the long-time simulations.
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