Discrete transparent boundary conditions for the mixed KDV-BBM equation

International audienceIn 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 holomor-phic 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

Numerical simulation of quantum waveguides

This chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano-scale semiconductor devices. We discuss transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a two dimensional domain. First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a compact nine-point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow

Discrete transparent boundary conditions for the Schrödinger equation on circular domains

We propose transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a circular computational domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson finite difference scheme. The presented discrete initial boundary-value problem is unconditionally stable and completely reflection-free at the boundary. Then, since the discrete TBCs for the Schrödinger equation with a spatially dependent potential include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate discrete TBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In numerical tests we finally illustrate the accuracy, stability, and efficiency of the proposed method. As a by-product we also present a new formulation of discrete TBCs for the 1D Schrödinger equation, with convolution coefficients that have better decay properties than those from the literature

Discrete transparent boundary conditions for the Schrödinger equation: Fast calculation, approximation, and stability

This paper is concerned with transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation in one and two dimensions. Discrete TBCs are introduced in the numerical simulations of whole space problems in order to reduce the computational domain to a finite region. Since the discrete TBC for the Schrödinger equation includes a convolution w.r.t. time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate TBCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove stability of the resulting initial-boundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples

Simulation of Laser Beam Propagation With a Paraxial Model in a Tilted Frame

We study the Schr\"odinger equation which comes from the paraxial approximation of the Helmholtz equation in the case where the direction of propagation is tilted with respect to the boundary of the domain. In a first part, a mathematical analysis is made which leads to an analytical formula of the solution in the simple case where the refraction index and the absorption coefficients are constant. Afterwards, we propose a numerical method for solving the initial problem which uses the previous analytical expression. Numerical results are presented. We also sketch an extension to a time dependant model which is relevant for laser plasma interaction

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present and compare several approaches to implement the classical transparent boundary condition into finite difference and finite element discretizations. We present in detail the approaches of the authors and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case

A Review of Artificial Boundary Conditions for the Schrödinger Equation

In this review we discuss techniques to solve numerically the time-dependent linear Schrödinger equation on unbounded domains. We present some recent approaches and describe alternative ideas pointing out the relations between these works

Artificial boundary conditions for the linearized Benjamin-Bona-Mahony equation

International audienceWe consider various approximations of artificial boundary conditions for linearized Benjamin-Bona-Mahoney equation. 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 derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our tranparent boundary conditions

Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x_0=0, x_1,x_2.... x_n} up to n steps where x_i represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Levy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek-Spitzer formula and the associated Sparre Andersen theorem.Comment: Lecture notes for the summer school "Fundamental Problems in Statistical Physics: XII" held at Leuven, Belgium (2009). 20 pages, 4 figures; typos corrected, a figure redrawn and new references and discussions adde

Subsequent female breast cancer risk associated with anthracycline chemotherapy for childhood cancer.

Anthracycline-based chemotherapy is associated with increased subsequent breast cancer (SBC) risk in female childhood cancer survivors, but the current evidence is insufficient to support early breast cancer screening recommendations for survivors treated with anthracyclines. In this study, we pooled individual patient data of 17,903 survivors from six well-established studies, of whom 782 (4.4%) developed a SBC, and analyzed dose-dependent effects of individual anthracycline agents on developing SBC and interactions with chest radiotherapy. A dose-dependent increased SBC risk was seen for doxorubicin (hazard ratio (HR) per 100 mg m-2: 1.24, 95% confidence interval (CI): 1.18-1.31), with more than twofold increased risk for survivors treated with ≥200 mg m-2 cumulative doxorubicin dose versus no doxorubicin (HR: 2.50 for 200-299 mg m-2, HR: 2.33 for 300-399 mg m-2 and HR: 2.78 for ≥400 mg m-2). For daunorubicin, the associations were not statistically significant. Epirubicin was associated with increased SBC risk (yes/no, HR: 3.25, 95% CI: 1.59-6.63). For patients treated with or without chest irradiation, HRs per 100 mg m-2 of doxorubicin were 1.11 (95% CI: 1.02-1.21) and 1.26 (95% CI: 1.17-1.36), respectively. Our findings support that early initiation of SBC surveillance may be reasonable for survivors who received ≥200 mg m-2 cumulative doxorubicin dose and should be considered in SBC surveillance guidelines for survivors and future treatment protocols