Transparent Boundary Conditions Research Articles

The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation

Abstract We consider the Cauchy problem for the 1D generalized Schrödinger equation on the whole axis. To solve it, any order finite element in space and the Crank–Nicolson in time method with the discrete transparent boundary conditions (TBCs) has recently been constructed. Now we engage the global Richardson extrapolation in time to derive the high order method both with respect to space and time steps. To study its properties, we comment on its stability and give results of numerical experiments and enlarged practical error analysis for three typical examples. Unlike most common second order (in either space or time step) methods, the proposed method is able to provide high precision results in the uniform norm by using adequate computational costs. It works even in the case of discontinuous potentials and non-smooth solutions far beyond the scope of its standard theory. Comparing our results to the previous ones, we obtain much more accurate results using much less amount of both elements and time steps.

On Mayfield’s stability proof for the discretized transparent boundary condition for the parabolic equation

Mayfield’s numerical implementation of transparent boundary condition for the Schrödinger-type parabolic equations is revisited. An inaccuracy in the original proof of the conditional stability for the resulting scheme is pointed out. The highly unusual and impressive original result is reestablished and a new proof is presented. Some further remarks and estimates on the instability which occurs when the Mayfield condition is violated are given.

An Adaptive Finite Element Method for the Diffraction Grating Problem with Transparent Boundary Condition

The diffraction grating problem is modeled by a boundary value problem governed by a Helmholtz equation with transparent boundary conditions. An a posteriori error estimate is derived when the truncation of the nonlocal boundary operators takes place. To overcome the difficulty caused by the fact that the truncated Dirichlet-to-Neumann (DtN) mapping does not converge to the original DtN mapping in its operator norm, a duality argument without assuming more regularity than the weak solution is applied. The a posteriori error estimate consists of two parts, the finite element discretization error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter. Based on the a posteriori error control, a finite element adaptive strategy is established for the diffraction grating problem, such that the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive algorithm.

Finite-time stabilization of a network of strings

We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term $\alpha u_t$ with a coefficient $\alpha$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $\alpha$, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node. Our results lead to the finite-time stabilization even though the systems may not be dissipative.

The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis

We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis.The finite-difference scheme with the Numerov averages on the non-uniform space mesh and of the Crank-Nicolson type in time is studied, with some approximate transparent boundary conditions (TBCs).Deriving bounds for the skew-Hermitian parts of the Numerov sesquilinear forms, we prove the uniform in time stability in $L^2$- and $H^1$-like space norms under suitable conditions on the potential and the meshes.In the case of the discrete TBC, we also derive higher order in space error estimates in both norms in dependence with the Sobolev regularity of the initial function (and the potential) and properties of the space mesh.Numerical results are presented for tunneling through smooth and rectangular potentials-wells, including the global Richardson extrapolation in time to ensure higher order in time as well.

Analysis of Transient Electromagnetic Scattering from a Three-Dimensional Open Cavity

This paper is concerned with the mathematical analysis of the time-domain Maxwell equations in a three-dimensional open cavity. An exact transparent boundary condition is developed to reformulate the open cavity scattering problem in an unbounded domain, equivalently, into an initial-boundary value problem in a bounded domain. The well-posedness and stability are studied for the reduced problem. Moreover, an a priori estimate is established for the electric field with a minimum regularity requirement for the data. (The PDF has been changed.)

Transparent boundary condition for numerical modeling of intense laser–molecule interaction

This article addresses the problem of constructing transparent boundary conditions for the Schrödinger–Maxwell system where the interacting electromagnetic field is treated classically in the dipole approximation. The problem essentially boils down to constructing transparent boundary condition for the Schrödinger equation with time-dependent linear potential. The transparent boundary maps obtained are not purely of the convolution-type; however, they are amenable to numerical methods like convolution quadrature for implementation.

A posterior error estimates for the nonlinear grating problem with transparent boundary condition

The nonlinear grating problem is modeled by Maxwell's equations with transparent boundary conditions. The nonlocal boundary operators are truncated by taking sufficiently many terms in the corresponding expansions. A finite element method with the truncation operators is developed for solving the nonlinear grating problem. The two posterior error estimates are established. The a posterior error estimate consists of two parts: finite element discretization error and the truncation error of the nonlocal boundary operators. In particular, the truncation error caused by truncation operations is exponentially decayed when the parameter N is increased. Numerical experiment is included to illustrate the efficiency of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1101–1118, 2015

Existence of solution of a non-linear multiband kp model with transparent boundary conditions

The existence of the solution of a multiband kp-model that describes the quantum mechanical motion of a system of particles in a crystal is discussed. The model includes the non-parabolicity effects of the high energetic particles in the valence band and the electrostatic potential of the charges. We consider a bounded domain and the ohmic contacts are modeled by using transparent boundary conditions. We include some numerical tests that illustrate the behavior of the solution in the proximity of the resonant states. This point constitutes one of the major difficulties of the mathematical analysis of the system.

Well-posedness of the Stokes–Coriolis system in the half-space over a rough surface

This paper is devoted to the well-posedness of the stationary $3$d Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of G\'erard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes-Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes-Coriolis system with data in the Kato space $H^{1/2}_{uloc}$.

Radar cross section reduction of a cavity in the ground plane: TE polarization

The reduction of backscatter radar cross section(RCS) in TE polarization for a rectangular cavity embedded in the ground plane is investigated in this paper. It is established by placing a thin, multilayered radar absorbing material(RAM) with possibly different permittivities at the bottom of the cavity. A minimization problem with respect to the backscatter RCS is formulated to determine the synthesis of RAM. The underlying scattered field is governed by a generalized Helmholtz equation with transparent boundary condition. The gradient with respect to the material permittivity is derived by the adjoint state method. A fast solver for the Helmholtz equation is presented for the optimization scheme. Numerical examples are presented to show the efficiency of the algorithm for RCS reduction.

The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation on a semi-innite strip. For the Crank-Nicolson nite-dierence scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniform in time L 2 -stability is proved. Due to the splitting, an eective direct algorithm using FFT is developed to implement the method with the discrete TBC for general potential. Numerical results on the tunnel eect for rectangular barriers are included together with the related practical error analysis. MSC classication: 65M06, 65M12, 35Q40

Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides

The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.

Reflectionless Beam Propagation on a Piecewise Linear Complex Domain

We describe a new class of transparent boundary conditions for beam-propagation simulation. The method is based on a transformation that breaks the original real-axis computational domain into a piecewise linear complex-plane contour, with higher-order analogues of Riemann-Cauchy conditions imposed at sharp corners of the contour. The resulting boundary conditions are extremely effective in terminating outgoing waves, and are easy to implement in an arbitrary beam-propagation method.

On a splitting higher-order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped

An initial-boundary value problem for the n-dimensional (n⩾2) time-dependent Schrödinger equation in a semi-infinite parallelepiped is considered. Starting from the Numerov–Crank–Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for n⩾3. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting double-splitting method, the uniqueness of solution and the uniform in time L2-stability are proved and an error estimate is stated. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applied to implement the scheme for general potential.

A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78–95, 2013) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier–Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem, 2013) and study numerically the minimization problem at low and mid-high frequencies. Numerical examples in some relevant cases are also shown.

Perfectly Matched Layers versus discrete transparent boundary conditions in quantum device simulations

Discrete transparent boundary conditions (DTBC) and the Perfectly Matched Layers (PML) method for the realization of open boundary conditions in quantum device simulations are compared, based on the stationary and time-dependent Schrödinger equation. The comparison includes scattering state, wave packet, and transient scattering state simulations in one and two space dimensions. The Schrödinger equation is discretized by a second-order Crank–Nicolson method in case of DTBC. For the discretization with PML, symmetric second-, fourth-, and sixth-order spatial approximations as well as Crank–Nicolson and classical Runge–Kutta time-integration methods are employed. In two space dimensions, a ring-shaped quantum waveguide device is simulated in the stationary and transient regime. As an application, a simulation of the Aharonov–Bohm effect in this device is performed, showing the excitation of bound states localized in the ring region. The numerical experiments show that the results obtained from PML are comparable to those obtained using DTBC, while keeping the high numerical efficiency and flexibility as well as the ease of implementation of the former method.

Application of transparent boundary conditions to high-order finite-difference schemes for the wave equation in waveguides

We propose a method for generating finite-difference approximations of transparent boundary conditions (TBCs) with the fourth and sixth order in space. It is based on the wave equation solution continuation extra two or three layers of grid points outside the computational domain to use them in central-difference operators on approaching the boundary. We present the theoretical background of the method, give estimates of computational resources, and discuss accuracy and stability results of numerical tests in 1D and 2D cases.

On a Numerov–Crank–Nicolson–Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov–Crank–Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L2-stability (in particular, L2-conservativeness) together with the error estimate O(τ2+|h|4) are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip and solving of tridiagonal systems in its main direction is developed to implement the splitting method for general potential. We also engage the Richardson extrapolation in time to increase the error order with respect to time step and get the method of higher order both in space and time. Numerical results on the tunnel effect for smooth and discontinuous rectangular barriers are included together with the careful practical error analysis on refining meshes.

A study of one-dimensional transport of Bose–Einstein condensates using exterior complex scaling

We numerically investigate the one-dimensional transport of Bose–Einstein condensates in the context of guided atom lasers using a mean-field description of the condensate in terms of a spatially discretized Gross–Pitaevskii equation. We specifically consider a waveguide configuration in which spatial inhomogeneities and nonvanishing atom–atom interactions are restricted to a spatially localized scattering region of finite extent. We show how the method of smooth exterior complex scaling can be implemented for this particular configuration in order to efficiently absorb the outgoing flux within the waveguide. A numerical comparison with the introduction of a complex absorbing potential as well as with the analytically exact elimination of the dynamics of the free non-interacting motion outside the scattering region, giving rise to transparent boundary conditions, clearly confirms the accuracy and efficiency of the smooth exterior complex scaling method.