Discrete Transparent Boundary Conditions Research Articles

Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

Absorbing boundary conditions are important when one simulates the propagation of waves on a bounded numerical domain without creating artificial reflections. In this paper, we consider various hyperbolic–dispersive equations modeling water wave propagation. A typical example is the Korteweg–de Vries equation 1 $$\begin{aligned} \displaystyle u_t+u\,u_x+\varepsilon u_{xxx}=0,\quad \forall x\in {\mathbb {R}}, \quad \forall t>0. \end{aligned}$$ In the case of linearized equations, some progress was recently done for one dimensional scalar dispersive equations using discrete transparent boundary conditions. However, a generalization of this approach to multi-dimensional setting is not obvious. In this paper, we consider the alternative perfectly matched layer (PML) approach for the linearized Korteweg–de Vries equation: 2 $$\begin{aligned} \displaystyle u_t+U\,u_x+\varepsilon u_{xxx}=0\quad \forall x\in {\mathbb {R}}, \quad \forall t>0, \end{aligned}$$ where $$U\in {\mathbb {R}}$$ denotes a reference speed. We first propose a direct perfectly matched layer approach and study the stability of the modified system. These equations are not always stable, the main obstruction being the classical condition $$v_g(k)v_\phi (k)\ge 0$$ found in the literature on PML (Bécache et al in J Comput Phys 188(2):399–433, 2003) that we recover in our analysis. Then, we introduce a hyperbolic system with a source term that is an approximation of the Korteweg–de Vries equations. In this case, the complete PML equations are not, again, completely stable. However, a version of the PML equations for this system derived without the source term is found to be stable and can absorb outgoing waves although it may create reflections as it is not perfectly matched. Finally, we consider the BBM–Boussinesq system that models bi-directional waves at the surface of an inviscid fluid layer. The dispersive properties for a subclass of physically relevant models are better suited for PML techniques since the condition $$v_g(k)v_\phi (k)\ge 0$$ is always satisfied. We show that the PML equations are always stable in this case. We illustrate numerically the absorbing and stability properties of these PML models and provide also KdV type simulation by choosing properly initial data.

Discrete transparent boundary conditions for the two-dimensional leap-frog scheme: approximation and fast implementation

We develop a general strategy in order to implement approximate discrete transparent boundary conditions for finite difference approximations of the two-dimensional transport equation. The computational domain is a rectangle equipped with a Cartesian grid. For the two-dimensional leap-frog scheme, we explain why our strategy provides with explicit numerical boundary conditions on the four sides of the rectangle and why it does not require prescribing any condition at the four corners of the computational domain. The stability of the numerical boundary condition on each side of the rectangle is analyzed by means of the so-called normal mode analysis. Numerical investigations for the full problem on the rectangle show that strong instabilities may occur when coupling stable strategies on each side of the rectangle. Other coupling strategies yield promising results.

Numerov-Padé Scheme for the One-Way Helmholtz Equation in Tropospheric Radio-Wave Propagation

This letter is devoted to applying the higher order finite-difference approximations to the tropospheric radio-wave propagation problem. It is proposed to use the fourth-order approximation (Numerov discretization) of the transverse differential operator, and rational Pade approximations of the propagation operator. A discrete transparent nonlocal boundary condition that can take into account the modified refractive index of the troposphere has been constructed. An error analysis of the considered scheme with emphasis on tropospheric propagation has been carried out. A comparison with the existing finite-difference schemes and the split-step Fourier method has been performed. The effectiveness of the proposed approach has been shown.

Discrete transparent boundary conditions for the equation of rod transverse vibrations

Local perturbations of an infinitely long rod travel to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local with respect to time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation, similar to the Hermite - Padé approximation is used. Numerical experiments confirm the high “transparency” of these boundary conditions and determine the conditional stability regions for finite-difference scheme.

Discrete Transparent Boundary Conditions for the Linearized Green--Naghdi System of Equations

In this paper, we introduce artificial boundary conditions for the linearized Green--Naghdi system of equations. The derivation of such continuous (respectively, discrete) boundary conditions inclu...

Transparent boundary conditions for higher-order finite-difference schemes of the Schrödinger equation in (1+1)D

A systematic development of discrete transparent boundary conditions (TBCs) for higher-order finite-difference approximations to Schrödinger-type equations is formulated and tested numerically. The finite-range interaction scheme for the original wave function is transformed into an equivalent nearest-neighbour coupling scheme for unit cells represented by multi-component spinors. This nearest-neighbour scheme lends itself to an established approach: the single-cell translation matrix for the extended spinor is constructed in Z space and cast into Jordan form. TBCs are obtained by setting out-bound growing spinor contributions equal to zero. Specifically we consider unitary extensions to the Crank–Nicolson time-propagation scheme for the time-dependent Schrödinger equation in (1+1)D. Numerical examples are given for selected higher-order propagation schemes. Finally, a simplified formulation based on a single, dominant growth factor is discussed and tested numerically.

A Domain Decomposition Method for Linearized Boussinesq-Type Equations

In this paper, we derive discrete transparent boundary conditions for a class of linearized Boussinesq equations. These conditions happen to be non-local in time and we test numerically their accuracy with a Crank-Nicolson time-discretization on a staggered grid. We use the derived transparent boundary conditions as interface conditions in a domain decomposition method, where they become local in time. We analyze numerically their efficiency thanks to comparisons made with other interface conditions.

Transparent boundary conditions for the Bogoliubov–de Gennes equation in (1+1)D

Discrete transparent boundary conditions (TBCs) are presented for a Crank–Nicolson implementation of the (1+1)D Bogoliubov–de Gennes equation for s- and p-wave superconductors. The whole-space dynamics is reformulated as a finite-space problem augmented by transparent boundary conditions. The latter are derived for the underlying space–time grid under the assumption of constant coefficients (mass, electric potential, and gap function) in the exterior regions. The translation matrix for the spinor on the spatial grid is expressed in Z space. The condition for out-projection of (exponentially) growing wave contributions establishes the TBCs. Their inverse Z transform then are fed into the Crank–Nicolson scheme. Next to basic numerical tests, the scheme is illustrated for the example of Andreev reflection at normal-metal–superconductor interfaces and compared to analytic results, where available. Suppression factors for reflected waves at superconducting boundaries are determined for selected numerical examples.

Discrete transparent boundary conditions for the mixed KDV–BBM equation

In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg–de Vries (KDV) and Benjamin–Bona–Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data.

Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis

AbstractWe consider the generalized time-dependent Schrödinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step

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.

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.

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

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.

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.

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 dispersion and norm preserving finite difference scheme with transparent boundary conditions for the Dirac equation in ( 1+1)D

A finite difference scheme is presented for the Dirac equation in ( 1 + 1 ) D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary cond...

Discrete transparent boundary conditions for the time-dependent Schrödinger equation: an explicit formulation

Using an explicit propagation algorithm for the time-dependent Schrödinger equation, based on finite differencing and a split operator technique, we formulate a non-Markovian dynamics for a restricted spatial grid with transparent boundary conditions. Applications to free particle wavepacket propagation and to the dissociation dynamics of a laser driven Morse oscillator are given.

A dispersion and norm preserving finite difference scheme with transparent boundary conditions for the Dirac equation in [formula omitted

A finite difference scheme is presented for the Dirac equation in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs). Based on a space- and time-staggered leap-frog scheme it avoids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly.Considering boundary regions, each with a constant mass and potential term, the associated DTBCs are derived by first applying this finite difference scheme and then using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in l2 which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflections at the boundaries and enforce stability of the whole space–time scheme.An exactly preserved functional for the norm of the Dirac spinor on the staggered grid is presented. Simulations of Gaussian wave packets, leaving the computational domain without reflection, demonstrate the quality of the DTBCs numerically, as well as the importance of a faithful representation of the energy–momentum dispersion relation on a grid.

A Family of Finite-Difference Schemes with Discrete Transparent Boundary Conditions for a Parabolic Equation on the Half-Axis

Abstract. An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averages both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes by applying the method of reproducing functions. Results of numerical experiments are included as well.