Higher-order, multi-dimensional fast discretizations with exact fractional artificial boundary conditions for the time-fractional Schrödinger equation on unbounded domains

The time-fractional Schrödinger equation (TFSE) — the Schrödinger equation with the first-order time derivative replaced by a Caputo derivative of order $\alpha\in(0,1)$ — describes non-Markovian quantum dynamics with memory. Li, Zhang and Zhang (2018) reduced the one-dimensional TFSE on the whole line to a bounded interval using exact artificial boundary conditions (ABCs) that involve Caputo derivatives of order $\alpha/2$ at the artificial boundaries, discretized the reduced problem by the L1 formula and central differences with unconditional $L^2$ stability, and accelerated the Caputo history by a sum-of-exponentials (SOE) approximation whose tolerance $\epsilon$ must satisfy $\epsilon\le c\,\Delta t^{1/2}h$. The open problem left by this framework, which remains unresolved, is to complete it: construct and rigorously analyze discretizations of higher order than the L1 order $2-\alpha$ in time and second order in space that keep the exact fractional ABCs, unconditional stability, and near-linear fast history evaluation; extend the exact ABC derivation together with the fast treatment and full stability/convergence analysis to higher spatial dimensions; obtain sharp error estimates for solutions with the generic weak initial-time singularity using graded meshes; and replace the static SOE tolerance condition $\epsilon\le c\,\Delta t^{1/2}h$ by an adaptive, certified rule. Later progress on each ingredient — higher-order schemes, multi-dimensional TFSE, graded-mesh analysis, fast evaluation — was made on bounded domains with standard boundary conditions, while the ABC-based line remains at the L1 order in one space dimension; the ABC-coupled combination is open.

Background

The time-fractional Schrödinger equation (TFSE) replaces the first-order time derivative of the Schrödinger equation by a Caputo fractional derivative of order $\alpha\in(0,1)$. The Caputo derivative is defined by $_{0}^{C}D_{t}^{\alpha}u(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha}\,\partial_{s}u(s)\,ds$; it is a history integral over the whole past, which makes the dynamics non-Markovian and is used to model anomalous quantum transport with memory effects. On the whole line the equation reads

$$\mathrm{i}\,{}_{0}^{C}D_{t}^{\alpha}u(x,t)=-\partial_{x}^{2}u(x,t)+V(x)u(x,t),\qquad x\in\mathbb{R},\;0<t\le T,$$

with compactly supported complex-valued initial data $u(x,0)=\varphi(x)$ and a real potential $V$ that vanishes at spatial infinity.

To compute such a problem one truncates the unbounded domain to a bounded interval and must supply artificial boundary conditions (ABCs): conditions imposed on the two artificial boundary points that make the restricted initial-boundary-value problem equivalent to the original Cauchy problem, so that outgoing waves are absorbed without spurious reflection. For the integer-order Schrödinger equation, the exact (transparent) ABC involves a half-order time derivative at the boundary, and Sun (2006) analyzed difference schemes with such ABCs on infinite domains. For the TFSE, Li, Zhang and Zhang (2018) derived the exact ABCs by a Laplace-transform solution of the exterior problem: at each artificial boundary the spatial derivative of the solution is matched to a Caputo derivative of half the interior order, $\alpha/2$, multiplied by the complex phase factor $e^{-\pi\mathrm{i}/4}$. They then discretized the reduced problem with the L1 formula — the standard piecewise-linear interpolation approximation of the Caputo derivative on a uniform time grid, of accuracy $O(\Delta t^{2-\alpha})$ for smooth solutions — combined with second-order central differences in space, proved unconditional $L^2$ stability, and derived global error bounds in which the boundary discretization contributes terms such as $h^{3/2}$ and $\Delta t^{2-\alpha/2}/h^{1/2}$. Because the direct evaluation of the Caputo derivative costs $O(N)$ work per time step and $O(N)$ memory over $N$ steps, they also built a fast scheme in which the history kernel is approximated by a sum of exponentials (SOE), $\frac{1}{t^{\alpha+1}}\approx\sum_{i=1}^{N_{\mathrm{exp}}}\omega_{i}e^{-s_{i}t}$, so that the history is carried by $O(\log N)$ exponentially weighted recurrence variables; the error analysis requires the SOE tolerance $\epsilon$ to satisfy the compatibility condition $\epsilon\le c\,\Delta t^{1/2}h$ for the fast scheme to retain the accuracy of the direct one.

Three limitations of this framework define the open problem. First, the temporal accuracy is limited to the L1 order $2-\alpha$ — below second order for every $\alpha\in(0,1)$ — and the spatial accuracy to second order. Second, the construction is one-dimensional: the Laplace-transform derivation of the ABC and the discrete kernel recurrences rely on the explicit one-dimensional exterior solution. Third, the error estimates assume $u\in C^{4,2}_{x,t}$ up to $t=0$, whereas solutions of time-fractional PDEs generically exhibit a weak singularity at the initial time (typically $\partial_{t}u\sim t^{\alpha-1}$, so $u$ is not twice continuously differentiable in time at $t=0$); handling this requires graded meshes, i.e. temporally nonuniform meshes refined toward $t=0$ such as $t_{k}=T(k/N)^{\gamma}$, for which the uniform-grid convolution weights used in the energy arguments are no longer available. A practical fourth issue is that the SOE tolerance $\epsilon$ must be chosen by hand to satisfy the static condition $\epsilon\le c\,\Delta t^{1/2}h$, and no adaptive rule exists.

The subsequent literature developed each of these ingredients in isolation. Higher-order time discretizations for the TFSE exist — for example schemes based on the Alikhanov (L2-$1_{\sigma}$) formula, a three-point interpolation approximation of the Caputo derivative achieving second-order temporal accuracy, second-order implicit difference schemes, and space–time isogeometric methods — as do multi-dimensional TFSE schemes and graded-mesh error analyses, but all of them on bounded computational domains with standard boundary conditions. The ABC-based line itself, including a nonlinear TFSE version and a transfer to the time-fractional Klein–Gordon equation, has remained at the L1 temporal order in one space dimension. What is missing is the coupling: discretizations that are simultaneously higher-order, valid in several space dimensions, robust to the initial-time singularity, and fast, while still enforcing the exact fractional ABCs with proven stability and sharp convergence rates.

Problem Statement

Construct, and rigorously analyze, discretizations of the time-fractional Schrödinger equation $\mathrm{i}\,{}_{0}^{C}D_{t}^{\alpha}u=-\partial_{x}^{2}u+V(x)u$ with $0<\alpha<1$, a real potential that vanishes outside a bounded computational region, and compactly supported initial data, on unbounded spatial domains, within the exact artificial-boundary framework of Li, Zhang and Zhang (2018), addressing all of the following connected aspects:

1. Higher order. A temporal discretization of order strictly higher than the L1 order $2-\alpha$ and a spatial discretization of order strictly higher than two, with the half-order ($\alpha/2$) fractional artificial boundary operators discretized to matching accuracy, so that the scheme's proven overall convergence order exceeds that of the L1/central-difference scheme while mesh-ratio-free (unconditional) $L^2$ stability is preserved.

2. Fast history evaluation. The higher-order scheme must retain near-linear total computational cost and $O(\log N)$-type memory in the number of time steps $N$, via a sum-of-exponentials or equivalent compression of the Caputo history, with a proven error bound in which the compression tolerance $\epsilon$ appears explicitly.

3. Higher spatial dimensions. An extension of the exact ABC derivation, the fast temporal treatment, and the stability/convergence analysis to at least two spatial dimensions for a time-fractional Schrödinger equation on an unbounded domain, addressing the multi-dimensional exterior problem in which the one-dimensional Laplace-transform recurrences do not apply.

4. Weak initial regularity and tolerance control. Convergence rates that are sharp for solutions with the generic initial-time singularity (e.g. $\partial_{t}u\sim t^{\alpha-1}$), obtained with nonuniform/graded temporal meshes and covering the fractional boundary operators, together with an adaptive, certified rule for choosing the compression tolerance $\epsilon$ in relation to the mesh sizes, going beyond the static compatibility condition $\epsilon\le c\,\Delta t^{1/2}h$.

These four connected aspects are coordinated components of a single objective: a computationally efficient, provably accurate artificial-boundary framework for the time-fractional Schrödinger equation on unbounded domains. They interact through the exact fractional boundary structure — a higher-order interior scheme needs a matching higher-order boundary-operator discretization that preserves the discrete dissipation inequality; the history compression must survive the upgraded scheme; graded meshes change the convolution weights everywhere including at the boundary.

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