Energy-conserving methods for the nonlinear Schrödinger equation

DOI10.1016/j.amc.2017.04.018zbMath1426.65202OpenAlexW2612276368MaRDI QIDQ2422847

Felice Iavernaro, Luigi Brugnano, Luigi Barletti, G. Frasca Caccia

Publication date: 21 June 2019

Published in: Applied Mathematics and Computation (Search for Journal in Brave)

Full work available at URL: https://kar.kent.ac.uk/61702/1/Frasca-Caccia%20Schroedinger.pdf

zbMATH Keywords

nonlinear Schrödinger equationHamiltonian partial differential equationsHamiltonian boundary value methodsline integral methodsenergy-conserving methodsHBVMs

Mathematics Subject Classification ID

NLS equations (nonlinear Schrödinger equations) (35Q55) Numerical methods for initial value problems involving ordinary differential equations (65L05) Method of lines for initial value and initial-boundary value problems involving PDEs (65M20) Numerical methods for Hamiltonian systems including symplectic integrators (65P10) Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems (37M15)

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Uses Software

• BiM
• BiMD
• LIMbook

Cites Work

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• The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: the case of symplecticity
• A simple framework for the derivation and analysis of effective one-step methods for ODEs
• On the preservation of phase space structure under multisymplectic discretization
• A note on the efficient implementation of Hamiltonian BVMs
• Geometric numerical integration and Schrödinger equations
• Energy conservation issues in the numerical solution of the semilinear wave equation
• Blended implicit methods for the numerical solution of DAE problems
• Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. II: Abstract splitting
• Energy conserving local discontinuous Galerkin methods for the nonlinear Schrödinger equation with wave operator
• Splitting integrators for nonlinear Schrödinger equations over long times
• A method for the integration in time of certain partial differential equations
• Finite-difference solutions of a non-linear Schrödinger equation
• The numerical study of blowup with application to a nonlinear Schrödinger equation
• The BiM code for the numerical solution of ODEs.
• A novel numerical approach to simulating nonlinear Schrödinger equations with varying coeffi\-cients
• Blended implementation of block implicit methods for ODEs
• Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation
• On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms
• Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation
• Blended block BVMs (B\(_3\)VMs): A family of economical implicit methods for ODEs
• Efficient implementation of Gauss collocation and Hamiltonian boundary value methods
• Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs
• Analysis of Hamiltonian boundary value methods (HBVMs): A class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
• Line Integral Methods for Conservative Problems
• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• Information Transmission Using the Nonlinear Fourier Transform, Part I: Mathematical Tools
• On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations
• A multisymplectic variational integrator for the nonlinear Schrödinger equation
• Simulating Hamiltonian Dynamics
• Methods for the Numerical Solution of the Nonlinear Schroedinger Equation
• Split-Step Methods for the Solution of the Nonlinear Schrödinger Equation
• Energy- and Quadratic Invariants--Preserving Integrators Based upon Gauss Collocation Formulae
• Numerical methods for Hamiltonian PDEs
• Geometric Numerical Integration
• Hamiltonian BVMs (HBVMs): A Family of “Drift Free” Methods for Integrating polynomial Hamiltonian problems
• Symplectic methods for the nonlinear Schrödinger equation
• Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity
• Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation
• Geometric integrators for the nonlinear Schrödinger equation

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