Functionally Fitted Energy-Preserving Methods for Solving Oscillatory Nonlinear Hamiltonian Systems

DOI10.1137/15M1032752zbMath1342.65231arXiv2012.13066MaRDI QIDQ5739965

Yuwen Li, Xin-Yuan Wu

Publication date: 7 July 2016

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

Full work available at URL: https://arxiv.org/abs/2012.13066

zbMATH Keywords

numerical resultsnonlinear Schrödinger equationoscillatory Hamiltonian systemsenergy-preserving methodscontinuous finite element methodsfunctionally fitted methodscontinuous-stage Runge-Kutta methods

Mathematics Subject Classification ID

Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations (65L60) 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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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• A derivation of energy-preserving exponentially-fitted integrators for Poisson systems
• A simple framework for the derivation and analysis of effective one-step methods for ODEs
• An energy-preserving exponentially-fitted continuous stage Runge-Kutta method for Hamiltonian systems
• Symmetric and symplectic exponentially fitted Runge-Kutta methods of high order
• Energy-preserving integrators and the structure of B-series
• Does variable step size ruin a symplectic integrator?
• A fourth-order symplectic exponentially fitted integrator
• Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems
• Extended RKN-type methods for numerical integration of perturbed oscillators
• Continuous finite element methods which preserve energy properties for nonlinear problems
• On functionally-fitted Runge-Kutta methods
• Structure preservation of exponentially fitted Runge-Kutta methods
• An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions
• Variable time step integration with symplectic methods
• A new look at finite elements in time: A variational interpretation of Runge-Kutta methods
• Exponential fitted Runge-Kutta methods of collocation type: Fixed or variable knot points?
• Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods
• A new high precision energy-preserving integrator for system of oscillatory second-order differential equations
• Inherently energy conserving time finite elements for classical mechanics
• Time finite element methods: a unified framework for numerical discretizations of ODEs
• On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages
• Preserving energy resp. dissipation in numerical PDEs using the ``Average Vector Field method
• Numerical integration of ordinary differential equations based on trigonometric polynomials
• Numerical integration of products of Fourier and ordinary polynomials
• Time integration and discrete Hamiltonian systems
• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• Water waves, nonlinear Schrödinger equations and their solutions
• Geometric integration using discrete gradients
• Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations
• P-stability and exponential-fitting methods for y = f(x,y)
• Energy- and Quadratic Invariants--Preserving Integrators Based upon Gauss Collocation Formulae
• Energy-preserving Runge-Kutta methods
• Geometric Numerical Integration
• One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems
• A functional fitting Runge-Kutta method with variable coefficients
• Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation
• Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems