Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation

Besse, Christophe; Ehrhardt, Matthias; Lacroix–Violet, Ingrid

doi:10.1002/num.22058

Cited by 18 publications

(59 citation statements)

References 27 publications

(91 reference statements)

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“…In this section we derive discrete artificial boundary conditions for the linearized Green-Naghdi system (2). In order to build up these conditions, we follow the strategy found in [5] and [6] and consider directly the problem on the fully discretized equations. In this section, we focus on spatial discretization on a staggered grid and Crank Nicolson time discretization.…”

Section: Discrete Transparent Boundary Conditions: Staggered Gridmentioning

confidence: 99%

“…Theorem 4.1. Let η, w be a smooth solution (2) and (5). We define the Z−transform of f (·, x) for all x ∈ [x , x r ] by…”

Section: Consistency Theoremmentioning

confidence: 99%

“…See e.g. [3], [4], for quantum evolution equations and [5] for an application in the case of the linearized (KdV) equation. The paper is organised as follow.…”

Section: Introductionmentioning

confidence: 99%

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

Kazakova¹,

Noble²

2020

SIAM J. Numer. Anal.

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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 include the inversion of Laplace transform (respectively Z-transform) and these boundary conditions are in turn non local in time. In the case of continuous boundary conditions, the inversion is done explicitly. We consider two spatial discretisations of the initial system either on a staggered grid or on a collocated grids, both of interest from the practical point of view. We use a Crank Nicolson time discretization. The proposed numerical scheme with the staggered grid permits explicit Z-transform inversion whereas the collocated grid discretization do not. A stable numerical procedure is proposed for this latter inversion. We test numerically the accuracy of the described method with standard Gaussian initial data and wave packet initial data which are more convenient to explore the dispersive properties of the initial set of equations. We used our transparent boundary conditions to solve numerically the problem of injecting propagating (planar) waves in a computational domain.

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Section: Discrete Transparent Boundary Conditions: Staggered Gridmentioning

confidence: 99%

“…Theorem 4.1. Let η, w be a smooth solution (2) and (5). We define the Z−transform of f (·, x) for all x ∈ [x , x r ] by…”

Section: Consistency Theoremmentioning

confidence: 99%

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

Kazakova¹,

Noble²

2020

SIAM J. Numer. Anal.

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“…It is therefore difficult to discretize the transparent boundary conditions (18) without any other knowledge. In [4] and [5], the construction of the discrete transparent boundary conditions for the approximation of the linearized Korteweg-de Vries equation (lKdV) (α = 0) and the linearized Benjamin-Bona-Mahoney equation (lBBM) (ε = 0) is made on fully discrete numerical schemes. In the case of the (lBBM) case, the space differential operator is of order two and it is possible to give explicit formulas both for the continuous and discrete transparent boundary conditions.…”

Section: Discrete Transparent Boundary Conditionsmentioning

confidence: 99%

“…This issue is also met at the discrete level where a numerical procedure is used to compute numerically the inverse Z transform. However, it requires an implementation with quadruple precision floating number in order to avoid instabilities as time becomes large (see [4] for more details). Here we propose an alternative approach to invert numerically the Z−transform which allows to construct "explicit" coefficient of discrete kernels.…”

Section: Discrete Transparent Boundary Conditionsmentioning

confidence: 99%

Discrete transparent boundary conditions for the mixed KDV–BBM equation

Besse

Noble

Sánchez

2017

Journal of Computational Physics

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In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) 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.

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Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

Trofimov

Loginova

Egorenkov

et al. 2023

Math Methods in App Sciences

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In this paper, we propose self‐adaptive artificial boundary conditions (ABCs) for the two‐dimensional (2D) time‐dependent nonlinear Schrödinger equation and studied their efficiency. This equation describes the laser pulse interaction with a semiconductor layer under laser‐induced plasma generation. Because of its nonlinearity, the interaction process is very sensitive to the appearance of the optical beam spurious reflection; therefore, highly transparent ABCs are urgently required to avoid disturbing the interaction process by an unphysical reflected wave. This can be achieved by using time‐ and spatial‐dependent local wavenumbers computed near the artificial boundaries. We claim that it is necessary to account for both wavenumbers in each of the adaptive ABCs stated on both spatial coordinates. Based on computer simulation, we demonstrate an increase in the efficiency of the adaptive ABC stated on the coordinate along which the optical pulse propagates by involving both local wavenumbers. The accuracy of the developed approach is verified using a well‐known analytical solution of the linear Schrödinger equation as well as the reference problem solution, obtained by solving the problem with zero‐value boundary conditions on the extended spatial domain. For computer modeling, the finite‐difference scheme, possessing asymptotic stability, is applied together with a two‐stage iterative process for its realization. The algorithm and peculiarities of the local wavenumber computation are discussed in detail. The applicability of the developed approach is also demonstrated via the solution of both linear and nonlinear problems. The proposed method has widespread application for solving various laser physics problems governed by the Schrödinger equation.

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Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation

Cited by 18 publications

References 27 publications

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

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

Discrete transparent boundary conditions for the mixed KDV–BBM equation

Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

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