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Calcolo
Published in: Calcolo 4/2017

01-12-2017

An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

Authors: Wei Shi, Kai Liu, Xinyuan Wu, Changying Liu

Published in: Calcolo | Issue 4/2017

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Abstract

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.
previous article A phase-field model for liquid–gas mixtures: mathematical modelling and discontinuous Galerkin discretization
next article A fully discrete scheme for the pressure–stress formulation of the time-domain fluid–structure interaction problem
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Metadata
Title
An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions
Authors
Wei Shi
Kai Liu
Xinyuan Wu
Changying Liu
Publication date
01-12-2017
Publisher
Springer Milan
Published in
Calcolo / Issue 4/2017
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-017-0232-5

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