Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation

DOI10.1016/j.cam.2018.10.014zbMath1414.65024OpenAlexW2899596537WikidataQ128997684 ScholiaQ128997684MaRDI QIDQ1715798

Gianmarco Gurioli, Luigi Brugnano, Ya-Juan Sun

Publication date: 29 January 2019

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

Full work available at URL: https://doi.org/10.1016/j.cam.2018.10.014

zbMATH Keywords

Korteweg-de Vries equationHamiltonian problemsHamiltonian partial differential equationsHamiltonian boundary value methodsenergy-conserving methodsHBVMs

Mathematics Subject Classification ID

PDEs in connection with fluid mechanics (35Q35) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs (65M70) Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems (37K40) Finite difference and finite volume methods for ordinary differential equations (65L12) Soliton solutions (35C08)

Related Items (14)

Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs ⋮ Spectrally accurate space-time solution of Manakov systems ⋮ Structure-preserving reduced-order modeling of Korteweg-de Vries equation ⋮ A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation ⋮ A second order convergent difference scheme for the initial‐boundary value problem of Korteweg–de Vires equation ⋮ Arbitrarily high‐order accurate and energy‐stable schemes for solving the conservative Allen–Cahn equation ⋮ Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation ⋮ A hybrid numerical method for the KdV equation by finite difference and sinc collocation method ⋮ A note on the continuous-stage Runge-Kutta(-Nyström) formulation of Hamiltonian boundary value methods (HBVMs) ⋮ Line integral solution of differential problems ⋮ Exponential collocation methods for conservative or dissipative systems ⋮ The exponential invariant energy quadratization approach for general multi-symplectic Hamiltonian PDEs ⋮ Spectrally accurate space-time solution of Hamiltonian PDEs ⋮ Unnamed Item

Uses Software

• Chebfun
• BiM
• Mulprec
• BiMD
• LIMbook

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• On decay properties of solutions of the \(k\)-generalized KdV equation
• A simple framework for the derivation and analysis of effective one-step methods for ODEs
• A note on the efficient implementation of Hamiltonian BVMs
• Energy conservation issues in the numerical solution of the semilinear wave equation
• A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation
• Blended implicit methods for the numerical solution of DAE problems
• Recent advances in linear analysis of convergence for splittings for solving ODE problems
• Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations
• Galerkin methods applied to some model equations for non-linear dispersive waves
• Accuracy and conservation properties in numerical integration: The case of the Korteweg-de Vries equation
• On the Korteweg-de Vries equation: Frequencies and initial value problem
• Multisymplectic box schemes and the Korteweg-de Vries equation.
• The BiM code for the numerical solution of ODEs.
• Blended implementation of block implicit methods for ODEs
• Boundary value methods: The third way between linear multistep and Runge-Kutta methods
• A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator
• Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation
• Energy preserving integration of bi-Hamiltonian partial differential equations
• An exponential-type integrator for the KdV equation
• Energy-conserving methods for the nonlinear Schrödinger equation
• Efficient implementation of Gauss collocation and Hamiltonian boundary value methods
• Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation
• A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect
• Analysis of Hamiltonian boundary value methods (HBVMs): A class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
• Some Analytic Results on the FPU Paradox
• Conservative, discontinuous Galerkin–methods for the generalized Korteweg–de Vries equation
• The KdV equation under periodic boundary conditions and its perturbations
• Line Integral Methods for Conservative Problems
• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
• On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces
• Local Structure-Preserving Algorithms for the KdV Equation
• Simulating Hamiltonian Dynamics
• Operator splitting for the KdV equation
• A Hamiltonian Approximation to Simulate Solitary Waves of the Korteweg-De Vries Equation
• On the Propagation of Regularity and Decay of Solutions to thek-Generalized Korteweg-de Vries Equation
• Numerical Methods in Scientific Computing, Volume I
• Fully discrete galerkin methods for the korteweg-de vries equation
• A Conservative Finite Element Method for the Korteweg-de Vries Equation
• Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation
• Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vries Equation as a Hamiltonian System
• A Local Discontinuous Galerkin Method for KdV Type Equations
• Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation
• On the well-posedness of the periodic KdV equation in high regularity classes
• A new class of energy-preserving numerical integration methods
• Geometric Numerical Integration
• Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension
• Hamiltonian BVMs (HBVMs): A Family of “Drift Free” Methods for Integrating polynomial Hamiltonian problems

This page was built for publication: Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation