Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schr\"odinger equation

TL;DR

This paper develops improved uniform error bounds for time-splitting methods applied to long-time Schrödinger equations with small potential and weak nonlinearity, introducing the RCO technique for better accuracy over extended times.

Contribution

The paper introduces the RCO technique to achieve sharper uniform error bounds for time-splitting methods in long-time Schrödinger dynamics with small potential and nonlinearity.

Findings

Established uniform error bounds of order C(T)(h^m + τ^2) for Schrödinger equations.

Proved improved error bounds of order O(h^{m-1} + ετ^2) in H^1-norm for long-time dynamics.

Extended RCO technique to nonlinear Schrödinger equations with enhanced error bounds.

Abstract

We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schr\"odinger equation with small potential and the nonlinear Schr\"odinger equation (NLSE) with weak nonlinearity. For the Schr\"odinger equation with small potential characterized by a dimensionless parameter $\varepsilon \in (0, 1]$ representing the amplitude of the potential, we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in $L^2$-norm to prove a uniform error bound at $C(T)(h^m +\tau^2)$ up to the long time $T_\varepsilon= T/\varepsilon$ for any $T>0$ and uniformly for $0<\varepsilon\le1$, while $h$ is the mesh size, $\tau$ is the time step, $m \ge 2$ depends on the regularity of the exact solution, and $C(T) =C_0+C_1T$ grows at most linearly with respect to $T$ with $C_0$ and $C_1$ two positive constants independent of $T$, $\varepsilon$, $h$ and $\tau$. Then by introducing a new technique of {\sl regularity compensation oscillation} (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform error bound at $O(h^{m-1} + \varepsilon \tau^2)$ is established in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon)$ of the Schr\"odinger equation with $O(\varepsilon)$-potential with $m \geq 3$, which is uniformly for $\varepsilon\in(0,1]$. Moreover, the RCO technique is extended to prove an improved uniform error bound at $O(h^{m-1} + \varepsilon^2\tau^2)$ in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon^2)$ of the cubic NLSE with $O(\varepsilon^2)$-nonlinearity strength, uniformly for $\varepsilon \in (0, 1]$. Extensions to the first-order and fourth-order time-splitting methods are discussed.