Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation
type: Item
instance of: scholarly article
title: Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation
zbMATH Open document ID: 1414.65024
author: Luigi Brugnano
author: Gianmarco Gurioli
author: Yajuan Sun
published in: Journal of Computational and Applied Mathematics
publication date: 2019-01-29T00:00:00Z
review text: The authors investigate the solutions of the well-known KdV equation in the case of periodic boundary conditions. To show the efficiency of the solutions, the authors apply a Fourier-Galerkin semidiscretization in space. This yields in a first step a Hamiltonian ODE problem. Next, in a second step, by observing an energy conservation in the Hamiltonian boundary value problem, the authors develop solutions in time by means of an energy-conserving method evolving Sylvester-type equations. The efficiency of the method is improved by the implementation of several numerical examples. \par Overall, it is an interesting work as it considers a famous problem of understanding the KdV equation which yields automatically an advance in understanding related problems such as wave propagation on the surface of shallow water, solitons, etc. \par The list of references is somehow exhaustive and reflects a good literature on the problem and the good background of the authors. \par I think finally that this work may open important directions by considering it on more complicated spaces such as fractal domains where a suitable notion of differentiablility, energy, periodicity should be defined.
reviewed by: Anouar Ben Mabrouk
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35C08
Mathematics Subject Classification ID: 37K40
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L12
zbMATH DE Number: 7007561
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: Hamiltonian partial differential equations
zbMATH Keywords: Hamiltonian problems
zbMATH Keywords: energy-conserving methods
zbMATH Keywords: Hamiltonian boundary value methods
zbMATH Keywords: HBVMs
describes a project that uses: LIMbook
describes a project that uses: BiM
describes a project that uses: BiMD
describes a project that uses: Chebfun
describes a project that uses: Mulprec
MaRDI profile type: Publication
full work available at URL: j.cam.2018.10.014
OpenAlex ID: W2899596537
cites work: On the well-posedness of the periodic KdV equation in high regularity classes
cites work: Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
cites work: Some Analytic Results on the FPU Paradox
cites work: Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vries Equation as a Hamiltonian System
cites work: On the Korteweg-de Vries equation: Frequencies and initial value problem
cites work: Q3706666
cites work: On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces
cites work: On decay properties of solutions of the \(k\)-generalized KdV equation
cites work: The KdV equation under periodic boundary conditions and its perturbations
cites work: On the Propagation of Regularity and Decay of Solutions to thek-Generalized Korteweg-de Vries Equation
cites work: Q4043472
cites work: Galerkin methods applied to some model equations for non-linear dispersive waves
cites work: A Conservative Finite Element Method for the Korteweg-de Vries Equation
cites work: Fully discrete galerkin methods for the korteweg-de vries equation
cites work: A Hamiltonian Approximation to Simulate Solitary Waves of the Korteweg-De Vries Equation
cites work: Accuracy and conservation properties in numerical integration: The case of the Korteweg-de Vries equation
cites work: A Local Discontinuous Galerkin Method for KdV Type Equations
cites work: A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect
cites work: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations
cites work: Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation
cites work: Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation
cites work: Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension
cites work: Multisymplectic box schemes and the Korteweg-de Vries equation.
cites work: Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation
cites work: A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator
cites work: Operator splitting for the KdV equation
cites work: An exponential-type integrator for the KdV equation
cites work: Local Structure-Preserving Algorithms for the KdV Equation
cites work: Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation
cites work: Energy preserving integration of bi-Hamiltonian partial differential equations
cites work: A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation
cites work: Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation
cites work: Q3151387
cites work: Simulating Hamiltonian Dynamics
cites work: Geometric Numerical Integration
cites work: Q4039908
cites work: Energy conservation issues in the numerical solution of the semilinear wave equation
cites work: Numerical Methods in Scientific Computing, Volume I
cites work: Q4315882
cites work: Line Integral Methods for Conservative Problems
cites work: Hamiltonian BVMs (HBVMs): A Family of “Drift Free” Methods for Integrating polynomial Hamiltonian problems
cites work: Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
cites work: A note on the efficient implementation of Hamiltonian BVMs
cites work: A simple framework for the derivation and analysis of effective one-step methods for ODEs
cites work: Efficient implementation of Gauss collocation and Hamiltonian boundary value methods
cites work: Analysis of Hamiltonian boundary value methods (HBVMs): A class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
cites work: Energy-conserving methods for the nonlinear Schrödinger equation

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (cites work)
cites work: A new class of energy-preserving numerical integration methods

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (cites work)
cites work: Blended implementation of block implicit methods for ODEs

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (cites work)
cites work: Boundary value methods: The third way between linear multistep and Runge-Kutta methods

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (cites work)
cites work: Recent advances in linear analysis of convergence for splittings for solving ODE problems

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (cites work)
cites work: The BiM code for the numerical solution of ODEs.

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (cites work)
cites work: Blended implicit methods for the numerical solution of DAE problems

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (cites work)
cites work: Q5689624

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (Wikidata QID)
Wikidata QID: Q128997684

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation (DOI)
DOI: 10.1016/J.CAM.2018.10.014