Transparent boundary condition and its effectively local approximation for the Schrödinger equation on a rectangular computational domain

TL;DR

The authors address the challenge of implementing transparent boundary conditions for the 2D Schrödinger equation on a rectangle by replacing the nonlocal boundary operator $\sqrt{\partial_t - i\triangle_{\Gamma}}$ with an effectively local Padé-based approximation. They develop two discretization pipelines, convolution-quadrature (CQ) and novel Padé (NP), with NP achieving time-step independent memory and computation by introducing auxiliary boundary fields governed by simple ODEs. A boundary-adapted Legendre-Galerkin spectral method is used to obtain a banded interior system, and a boundary lifting homogenizes Robin-type BCs. Numerical experiments with chirped-Gaussian and Hermite-Gaussian wave packets demonstrate accurate DtN maps, stable evolution, and convergence consistent with the underlying time-stepping schemes, validating the practicality and efficiency of NP over conventional Padé and CP methods. The approach is scalable to higher dimensions and promising for large-scale Schrödinger simulations where long-time integration would be prohibitive with history-dependent methods.

Abstract

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form $\sqrt{\partial_t-i\triangle_Γ}$ where $\triangle_Γ$ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

Figures (16)

• Figure 1: The figure shows a rectangular domain with boundary segments parallel to one of the axes.
• Figure 2: A schematic depiction of the evolution of the auxiliary field $\varphi(x_1,x_2,\tau_1,\tau_2)$ in the $(\tau_1,\tau_2)$-plane is provided in this figure where the plot on the right corresponds $\boldsymbol{x}\in\Gamma_r\cup\Gamma_l$ and the plot on the left corresponds $\boldsymbol{x}\in\Gamma_t\cup\Gamma_b$. The filled circles depict the evolution of the auxiliary field $\varphi(x_1,x_2,\tau_1,\tau_2)$ either above or below the diagonal in the $(\tau_1,\tau_2)$-plane starting from the diagonal which also serves as initial conditions for solving IVPs corresponding to the auxiliary function. The TBCs for the auxiliary field require the history of the auxiliary field at the corner points which makes empty circles relevant. Note that these values at the corners can be taken from the adjacent segment of the boundary where it is already being computed and this is depicted by broken lines. Note that the vertical/horizontal lines where the arrows end corresponds to the history of the auxiliary field needed for the TBCs on $\partial\Omega_i$ in the current time ($t$).
• Figure 3: A schematic depiction of the evolution of the auxiliary fields $\varphi_{k,a_1}(x_2,\tau_1,\tau_2),\; \varphi_{k,a_2}(x_1,\tau_1,\tau_2)$ and $\psi_{k,k',a_1,a_2}(\tau_1,\tau_2)$ in the $(\tau_1,\tau_2)$-plane is provided in this figure. The plots (A) and (B) depict the evolution of the fields $\varphi_{k,a_2}(x_1,\tau_1,\tau_2)$ and $\varphi_{k,a_1}(x_2,\tau_1,\tau_2)$ on the boundary segments $\Gamma_{a_2}$ and $\Gamma_{a_1}$, respectively. The plot (C) depicts the evolution of the field $\psi_{k,k',a_1,a_2}(\tau_1,\tau_2)$ which can be achieved by moving either below or above the diagonal.
• Figure 4: The figure shows the evolution of the relative energy content as defined in \ref{['eq:cg2d-energy-content']} of the chirped-Gaussian and the Hermite-Gaussian profiles considered in Table \ref{['tab:cg2d']} and Table \ref{['tab:hg2d']}. Here the computational domain is $\Omega_i=(-10,10)^2$.
• Figure 5: The figure shows the behaviour of error involved in the discretization of the boundary map alone for the chirped-Gaussian profile with different values of the speed 'c' (see Table \ref{['tab:cg2d']}). The numerical parameters and the labels are described in Sec. \ref{['sec:tests-mt']} where the error is quantified by \ref{['eq:error-dtn']}.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Definition 1: Riemann-Liouville fractional integrals
• Theorem 1: Law of exponents
• Definition 2: Riemann-Liouville fractional derivatives