Local discontinuous Galerkin methods for nonlinear Schrödinger equations

Authors:
Yan Xu;Chi-Wang Shu
Affiliations:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China;Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA
Venue:
Journal of Computational Physics
Year:
2005

Citing 15
Cited 21

Efficient implementation of essentially non-oscillatory shock-capturing schemes

Journal of Computational Physics
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems

Journal of Computational Physics
Pseudo-spectral solution of nonlinear Schro¨dinger equations

Journal of Computational Physics
A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations

Journal of Computational Physics
Classification of the solitary waves in coupled nonlinear Schro¨dinger equations

Physica D
A space-time finite element method for the nonlinear Schro¨dinger equation: the discontinuous Galerkin method

Mathematics of Computation
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

SIAM Journal on Numerical Analysis
Difference schemes for solving the generalized nonlinear Schrödinger equation

Journal of Computational Physics
A Space-Time Finite Element Method for the Nonlinear Schrödinger Equation: The Continuous Galerkin Method

SIAM Journal on Numerical Analysis
Solving the Generalized Nonlinear Schrödinger Equation via Quartic Spline Approximation

Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

Journal of Scientific Computing
Numerical simulation of coupled nonlinear Schrödinger equation

Mathematics and Computers in Simulation
Exponential time differencing for stiff systems

Journal of Computational Physics
A Local Discontinuous Galerkin Method for KdV Type Equations

SIAM Journal on Numerical Analysis
Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

Journal of Scientific Computing

Spectral/hp discontinuous Galerkin methods for modelling 2D Boussinesq equations

Journal of Computational Physics
A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect

Journal of Computational Physics
Local discontinuous Galerkin methods for the Cahn-Hilliard type equations

Journal of Computational Physics
A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions

Journal of Computational Physics
Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method

Mathematics and Computers in Simulation
A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger-Poisson equations with discontinuous potentials

Journal of Computational and Applied Mathematics
Lattice Boltzmann Simulation of Some Nonlinear Complex Equations

ICCS '07 Proceedings of the 7th international conference on Computational Science, Part I: ICCS 2007
A Schwarz generalized eigen-oscillation spectral element method (GeSEM) for 2-D high frequency electromagnetic scattering in dispersive inhomogeneous media

Journal of Computational Physics
Application of Discontinuous Galerkin Methods for Reaction-Diffusion Systems in Developmental Biology

Journal of Scientific Computing
Local Discontinuous Galerkin Method for Surface Diffusion and Willmore Flow of Graphs

Journal of Scientific Computing
Local discontinuous Galerkin methods for the generalized Zakharov system

Journal of Computational Physics
Symplectic wavelet collocation method for Hamiltonian wave equations

Journal of Computational Physics
A local discontinuous Galerkin method for directly solving Hamilton-Jacobi equations

Journal of Computational Physics
Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

Applied Numerical Mathematics
Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations

Applied Numerical Mathematics
The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs

Journal of Computational and Applied Mathematics
Optimal Error Estimates of the Semidiscrete Local Discontinuous Galerkin Methods for High Order Wave Equations

SIAM Journal on Numerical Analysis
A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system

Computers & Mathematics with Applications
Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods

Numerical Algorithms
A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: Energy conservation and boundary effect

Journal of Computational Physics
Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation

Calcolo: a quarterly on numerical analysis and theory of computation

Quantified Score

Hi-index	31.49

Visualization

Abstract

In this paper we develop a local discontinuous Galerkin method to solve the generalized nonlinear Schrodinger equation and the coupled nonlinear Schrodinger equation. L^2 stability of the schemes are obtained for both of these nonlinear equations. Numerical examples are shown to demonstrate the accuracy and capability of these methods.