An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions
type: Item
instance of: scholarly article
title: An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions
zbMATH Open document ID: 1395.65090
author: Wei Shi
author: Kai Liu
author: Xinyuan Wu
author: Changying Liu
published in: Calcolo
publication date: 2018-02-15T00:00:00Z
review text: As model problem the authors consider the nonlinear Klein-Gordon equation \[ \frac{\partial^2 u(x,t)}{\partial t^2} - \lambda \frac{\partial^2 u(x,t)}{\partial x^2} = - \nabla V(u) \, , \; (x,t) \in [x_L,x_R] \times [t_0,T], \] where \(V(u)\) is a smooth potential energy and \(\lambda\) is a positive real number. Additionally, initial conditions and Neumann boundary conditions are given. For the discretization in space the finite element method with standard continuous piecewise polynomial basis functions is used. This results in a system of ordinary differential equations. The stability of this semi-discrete finite element scheme is analysed. For the time discretization of the system of ordinary differential equations the average vector field approach is used. In this way the semi-discrete energy is preserved. The presented approach is illustrated in detail in the case of piecewise-linear basis functions in the finite element method. Finally, the conservative property of the presented algorithm is shown by two numerical examples.
reviewed by: Michael Jung
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 6839403
zbMATH Keywords: nonlinear Hamiltonian wave equation
zbMATH Keywords: energy-preserving schemes
zbMATH Keywords: finite element discretization
zbMATH Keywords: average vector field method
MaRDI profile type: Publication
full work available at URL: s10092-017-0232-5
OpenAlex ID: W2741427625
cites work: Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations
cites work: Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators
cites work: New travelling wave solutions to the Boussinesq and the Klein-Gordon equations
cites work: Q3961178
cites work: Soliton perturbation theory for phi-four model and nonlinear Klein-Gordon equations
cites work: Numerical solution of the Klein-Gordon equation via He's variational iteration method
cites work: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices
cites work: Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method
cites work: On the numerical solution of the Klein‐Gordon equation
cites work: Handbook of numerical analysis. Vol. XI. Special volume: Foundations of computational mathematics.
cites work: Finite Difference Calculus Invariant Structure of a Class of Algorithms for the Nonlinear Klein–Gordon Equation
cites work: Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method
cites work: Preserving energy resp. dissipation in numerical PDEs using the ``Average Vector Field'' method
cites work: Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: application to the vibrating piano string
cites work: Q4023226
cites work: A new class of energy-preserving numerical integration methods
cites work: Q3782682
cites work: Geometric integration using discrete gradients
cites work: Preserving algebraic invariants with Runge-Kutta methods
cites work: B-series methods cannot be volume-preserving
cites work: On energy conservation of the simplified Takahashi-Imada method
cites work: Linearization-preserving self-adjoint and symplectic integrators
cites work: Efficient energy-preserving integrators for oscillatory Hamiltonian systems
cites work: An extended discrete gradient formula for oscillatory Hamiltonian systems
cites work: Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations
cites work: Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations
cites work: Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems
cites work: Q5634306
cites work: Numerical Methods for Ordinary Differential Equations
cites work: Symmetric Multistip Methods for Periodic Initial Value Problems
cites work: Fourth-order compact solution of the nonlinear Klein-Gordon equation
cites work: A numerical method for the one-dimensional sine-Gordon equation
cites work: High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods
DOI: 10.1007/S10092-017-0232-5
description: scientific article; zbMATH DE number 6839403