A general framework for finding energy dissipative/conservative \(H^1\)-Galerkin schemes and their underlying \(H^1\)-weak forms for nonlinear evolution equations

DOI10.1007/s10543-014-0483-3zbMath1307.65137OpenAlexW1982213114MaRDI QIDQ486716

Yuto Miyatake, Takayasu Matsuo

Publication date: 16 January 2015

Published in: BIT (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10543-014-0483-3

zbMATH Keywords

Galerkin methoddissipationnonlinear evolution equationsSwift-Hohenberg equationCamassa-Holm equationconservationnumerical resultKawahara equationdiscrete gradient methodstructure-preserving integrationdiscrete partial derivative method\(L^2\)-projectionP1 elements

Mathematics Subject Classification ID

PDEs in connection with fluid mechanics (35Q35) Finite element methods applied to problems in fluid mechanics (76M10) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60)

Related Items

A fully discrete curve-shortening polygonal evolution law for moving boundary problems ⋮ Energy-preserving exponential integrators of arbitrarily high order for conservative or dissipative systems with highly oscillatory solutions ⋮ Energy-preserving \(H^1\)-Galerkin schemes for shallow water wave equations with peakon solutions ⋮ Conservative finite volume element schemes for the complex modified Korteweg-de Vries equation ⋮ Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs ⋮ A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations ⋮ Numerical integration of the Ostrovsky equation based on its geometric structures ⋮ A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity ⋮ Efficient mass- and energy-preserving schemes for the coupled nonlinear Schrödinger-Boussinesq system ⋮ Efficient energy-preserving wavelet collocation schemes for the coupled nonlinear Schrödinger-Boussinesq system ⋮ A local discontinuous Galerkin method based on variational structure

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• The discrete variational derivative method based on discrete differential forms
• Numerical integration of the Ostrovsky equation based on its geometric structures
• New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations
• Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations
• A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation
• An extension of the discrete variational method to nonuniform grids
• Symplectic structures, their Bäcklund transformations and hereditary symmetries
• An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations
• Hamiltonian-conserving discrete canonical equations based on variational difference quotients
• Finite difference schemes for \(\frac{\partial u}{\partial t}=(\frac{\partial}{\partial x})^\alpha\frac{\delta G}{\delta u}\) that inherit energy conservation or dissipation property
• Energy-preserving \(H^1\)-Galerkin schemes for shallow water wave equations with peakon solutions
• Preserving energy resp. dissipation in numerical PDEs using the ``Average Vector Field method
• Time integration and discrete Hamiltonian systems
• An alternating discrete variational derivative method for coupled partial differential equations
• Discrete Variational Derivative Method
• A General Framework for Deriving Integral Preserving Numerical Methods for PDEs
• Theoretical Numerical Analysis
• Numerical Analysis of a Continuum Model of Phase Transition
• Stability of peakons
• An integrable shallow water equation with peaked solitons
• A new class of energy-preserving numerical integration methods
• Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations