Discrete $H^1$ error estimates and higher-order-in-time boundary discretizations for the artificial-boundary finite difference scheme for nonlinear Schrödinger equations on unbounded domains

Wang and Liang (Appl. Numer. Math. 128, 2018) constructed a finite difference scheme for the one-dimensional cubic nonlinear Schrödinger equation on the real line, truncated by exact nonlocal artificial boundary conditions in which the boundary normal derivative is a time-convolution (fractional, half-order in time) operator on the boundary trace. Coupling a Crank–Nicolson interior discretization, an explicit extrapolation of the cubic term, and history-sum discretizations of the boundary fractional convolutions, they proved unconditional stability and an $L^2$-type error bound, first-order in time. The authors themselves state two open problems: deriving a discrete $H^1$-type error estimate for the nonlinear scheme, and constructing higher-order-in-time discretizations of the boundary history sums that retain the sign/positivity structure required for unconditional-stability energy estimates (an analogue of their Lemma 5).

Background

The nonlinear Schrödinger equation (NLS) describes dispersive wave phenomena such as soliton propagation in optical fibres, matter waves in Bose–Einstein condensates, and rogue-wave dynamics. In the normalization of the source paper it reads $i\,\partial_t\psi=-\tfrac12\,\partial_{xx}\psi+V(x,t)\,\psi+g(x,t)\,|\psi|^2\,\psi$ for $x\in\mathbb{R}$, where $V$ is an external potential and $g$ a (possibly space- and time-dependent) nonlinearity coefficient. Computing on the whole line requires truncating $\mathbb{R}$ to a finite interval $(x_L,x_R)$ and imposing artificial boundary conditions (ABCs) — boundary operators that reproduce the effect of the discarded exterior on the finite interval. When $V$ is constant in each exterior half-line and $g$ vanishes outside a bounded region, the exterior problem is linear and can be solved exactly by Laplace transforms, yielding Dirichlet-to-Neumann (DtN) maps: exact boundary conditions expressing $\partial_x\psi$ at each artificial boundary as a time-convolution operator with the singular kernel $(t-s)^{-1/2}$ acting on the boundary trace of $\psi$ (Zheng, Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, 2006). Such operators are nonlocal in time and of fractional (half) order in $t$, reflecting the dispersive character of the equation.

For the linear Schrödinger equation, Sun and Wu (J. Comput. Phys. 214/219, 2006) built finite difference schemes with these ABCs and proved unconditional stability and $L^2$-type convergence. Their key device is the discretization of the boundary time convolutions by history sums — discrete convolutions over all previous time levels with coefficients $a_k=2/(\sqrt{k}+\sqrt{k+1})$ — which approximate the fractional kernel and whose discrete convolution possesses a positivity (accretivity) property: the boundary history terms enter the discrete energy balance with a non-positive contribution, so the discrete energy never grows. Wang and Liang (2018) extended this to the nonlinear equation: on a staggered grid (unknowns sampled at half-integer mesh points) with the auxiliary variable $u=\partial_x\psi$, a Crank–Nicolson-type implicit interior discretization, and an explicit second-order extrapolation of the cubic term $g|\psi|^2\psi$ (avoiding nonlinear solves), they proved unconditional stability $\|\Psi^n\|_h\le\|\psi^0\|_h$ in a discrete $L^2$-type norm $\|v\|_h=(\sum_j|v_{j-1/2}|^2h)^{1/2}$ and a convergence bound in that same norm, giving first-order accuracy in time under the mesh coupling $\tau=O(h)$ (numerics show $O(\tau)$ in time and $O(h^2)$ in space). The analysis leaves two gaps, explicitly stated by the authors: (i) the error is controlled only in the discrete $L^2$-type norm — no discrete $H^1$-type (gradient) error estimate exists for the nonlinear scheme; (ii) the scheme is first-order in time, because no higher-order history-sum quadrature with the required sign/positivity structure (an analogue of the paper's Lemma 5) was available.

Neighboring literatures reach related goals by different routes. Local (Padé-type) ABCs approximate the exterior DtN operator by rational functions of $\partial_t$ expanded around a chosen reference wavenumber; with such boundary conditions, unconditionally stable second-order schemes with rigorous error bounds exist for the linear Schrödinger equation (Li, Zhang and Zheng, SIAM J. Numer. Anal., 2018; Xie and Li, Appl. Numer. Math., 2023, using BDF2, i.e. a two-step second-order backward differentiation formula), and Pan, Xia, He and Zhang (Appl. Numer. Math., 2020) proved unconditional stability and second-order accuracy in time and space for a three-level linearized difference scheme for the cubic NLS with third-order local ABCs; in all these cases the boundary conditions themselves are approximate, and their exterior modeling error is not controlled by the mesh. The discrete transparent boundary condition (DTBC) school derives the boundary condition from the discrete exterior problem itself (via the $z$-transform, a generating-function transform for discrete sequences), which preserves stability by construction: for the linear generalized Schrödinger equation this yields second-order-in-time schemes with full error estimates, including mesh $H^1$-norm estimates (Zlotnik, 2015; Ducomet, Zlotnik and Romanova, 2015), and Wang, Zhang and Zheng (Math. Comp. 91, 2022) proved stability and convergence of a second-order Crank–Nicolson scheme with discrete DtN-type boundary conditions for the nonlocal (PT-symmetric) NLS. On bounded domains, where no exterior problem arises, $H^1$-norm and superconvergence error analyses for cubic-NLS finite element methods are standard (e.g. Yang, Jia and Yang, 2025). None of these results provides an $H^1$-type error bound, or a higher-order-in-time discretization, of the exact nonlocal fractional DtN boundary operators for the local cubic NLS in the Wang–Liang scheme family.

Problem Statement

Consider the one-dimensional cubic nonlinear Schrödinger equation on the whole real line, in the normalization of Wang and Liang (2018),

$$i\,\partial_t\psi(x,t)=-\tfrac12\,\partial_{xx}\psi(x,t)+V(x,t)\,\psi(x,t)+g(x,t)\,|\psi(x,t)|^2\,\psi(x,t),\qquad x\in\mathbb{R},\ \ t\in(0,T],$$

with initial data $\psi(x,0)=\psi^0(x)$, under the structural assumptions that the potential is constant in the exterior, $V(x,t)\equiv V_l$ for $x\le x_L$ and $V(x,t)\equiv V_r$ for $x\ge x_R$, and that the nonlinearity is compactly supported, $g(x,t)=0$ outside a bounded interior region. Truncating to $\Omega_{in}=(x_L,x_R)$ and solving the linear exterior problems exactly yields the nonlocal artificial boundary conditions (in the notation of the paper)

$$\frac{\partial\psi(x_L,t)}{\partial x}=\sqrt{\tfrac{2}{\pi}}\,e^{-i(\pi/4+V_l t)}\,\frac{d}{dt}\int_0^t\frac{\psi(x_L,s)\,e^{iV_l s}}{\sqrt{t-s}}\,ds,\qquad \frac{\partial\psi(x_R,t)}{\partial x}=-\sqrt{\tfrac{2}{\pi}}\,e^{-i(\pi/4+V_r t)}\,\frac{d}{dt}\int_0^t\frac{\psi(x_R,s)\,e^{iV_r s}}{\sqrt{t-s}}\,ds.$$

Wang and Liang discretize this truncated problem by a Crank–Nicolson-type interior scheme on a staggered grid with the auxiliary variable $u=\partial_x\psi$, an explicit extrapolation of the cubic term, and history-sum discretizations of the two boundary fractional convolutions with coefficients $a_k=2/(\sqrt{k}+\sqrt{k+1})$ that are exact in space; they proved unconditional stability and an $L^2$-type error bound for this scheme. Two questions, stated as open in that paper, remain:

1. ($H^1$ error estimate) Prove an error bound in a discrete $H^1$-type norm — one controlling the discrete spatial derivative of the error (for example the discrete gradient on the staggered mesh, together with the error in the auxiliary variable $u=\partial_x\psi$) — for the Wang–Liang scheme, with explicit dependence of the bound on the mesh sizes $h$ and $\tau$, the final time $T$, and the assumed regularity of the exact solution, and identify the convergence rate in that norm.

2. (Higher-order-in-time boundary discretizations) Construct a discretization of the boundary fractional time-convolution operators — a higher-order analogue of the history sums $a_k=2/(\sqrt{k}+\sqrt{k+1})$ — that simultaneously (a) approximates the singular kernels $(t-s)^{-1/2}$ to at least second order in the time step, under regularity assumptions on the boundary traces compatible with the singularity of the kernel, and (b) provably satisfies the sign/positivity (accretivity) structure required by discrete energy estimates: as for the coefficients $a_k=2/(\sqrt{k}+\sqrt{k+1})$, the boundary history terms enter the discrete energy balance with a non-positive contribution, so that the discrete energy does not grow — the missing analogue of the paper's Lemma 5. Such a boundary discretization is the obstruction, identified in the source paper, to raising the temporal order of the scheme: coupling it with a matching at-least-second-order interior time discretization of the cubic nonlinearity would then give a fully discrete scheme of the corresponding order, amenable to the unconditional-stability energy argument; the open question is the construction of a boundary quadrature satisfying (a) and (b).

These two questions form one program: completing the convergence theory of unconditionally stable finite difference schemes that discretize the exact nonlocal artificial boundary conditions for the cubic nonlinear Schrödinger equation on unbounded domains — sharpening the norm of the existing theory (question 1) and its temporal order (question 2). They may be answered separately.

References

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