An Efficient Second-Order Finite Difference Method for the One-Dimensional Schrödinger Equation with Absorbing Boundary Conditions

DOI10.1137/17M1122347zbMath1450.65079OpenAlexW2795150153MaRDI QIDQ4635516

Jiwei Zhang, Buyang Li, Chunxiong Zheng

Publication date: 23 April 2018

Published in: SIAM Journal on Numerical Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/17m1122347

zbMATH Keywords

absorbing boundary conditionPadé approximationSchrödinger equationerror estimatefast algorithmconvolution quadrature

Mathematics Subject Classification ID

Numerical methods for integral equations (65R20) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Padé approximation (41A21) Error bounds for initial value and initial-boundary value problems involving PDEs (65M15) Applications to the sciences (65Z05) Time-dependent Schrödinger equations and Dirac equations (35Q41)

Related Items (11)

A fast second-order discretization scheme for the linearized Green-Naghdi system with absorbing boundary conditions ⋮ A fast stable accurate artificial boundary condition for the linearized Green-Naghdi system ⋮ Stability and Error Analysis for a Second-Order Fast Approximation of the One-dimensional Schrödinger Equation Under Absorbing Boundary Conditions ⋮ A fast BDF2 Galerkin finite element method for the one-dimensional time-dependent Schrödinger equation with artificial boundary conditions ⋮ A fast accurate artificial boundary condition for the Euler-Bernoulli beam ⋮ Two-grid algorithm for two-dimensional linear Schrödinger equation by the mixed finite element method ⋮ Fast artificial boundary method for the heat equation on unbounded domains with strip tails ⋮ Towards perfectly matched layers for time-dependent space fractional PDEs ⋮ Stability and Error Analysis for a Second-Order Fast Approximation of the Local and Nonlocal Diffusion Equations on the Real Line ⋮ Numerical computations of nonlocal Schrödinger equations on the real line ⋮ Stability and error analysis for a second-order approximation of 1D nonlocal Schrödinger equation under DtN-type boundary conditions

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Nonreflecting boundary conditions for the time-dependent wave equation
• ``Free-space boundary conditions for the time dependent wave equation
• Non-reflecting boundary conditions
• Exact artifical boundary conditions for the Schrödinger equation in \(\mathbb R ^2\)
• An analysis of the finite-difference method for one-dimensional Klein-Gordon equation on unbounded domain
• Convolution quadrature and discretized operational calculus. I
• Wave propagation and underwater acoustics
• Numerical solution of problems on unbounded domains. A review
• Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation.
• Exact and approximating boundary conditions for the parabolic problems on unbounded domains
• Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension
• Convolution quadrature revisited
• Discrete transparent boundary conditions for the numerical solution of Fresnel's equation
• Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation
• High-order local non-reflecting boundary conditions: a review
• Absorbing boundary conditions for the multidimensional Klein-Gordon equation
• The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions
• Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability
• MATHEMATICAL CONCEPTS OF OPEN QUANTUM BOUNDARY CONDITIONS
• Wave Propagation Problems Treated with Convolution Quadrature and BEM
• Absorbing Boundary Conditions for General Nonlinear Schrödinger Equations
• Fast Numerical Solution of Nonlinear Volterra Convolution Equations
• A Padé Approximation Method for Square Roots of Symmetric Positive Definite Matrices
• Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation
• Analysis of High-Order Absorbing Boundary Conditions for the Schrödinger Equation
• Design of Absorbing Boundary Conditions for Schrödinger Equations in $\mathbbR$^d
• Fast Convolution for Nonreflecting Boundary Conditions
• Artificial Boundary Method
• Exact Nonreflecting Boundary Condition For Elastic Waves
• Numerical solution to a linearized KdV equation on unbounded domain
• Computation of the Schrödinger Equation in the Semiclassical Regime on an Unbounded Domain
• Nonreflecting boundary condition for the wave equation

This page was built for publication: An Efficient Second-Order Finite Difference Method for the One-Dimensional Schrödinger Equation with Absorbing Boundary Conditions