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Linearly implicit local and global energy-preserving methods for PDEs with a cubic Hamiltonian
[article]
2020
arXiv
pre-print
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We present linearly implicit methods that preserve discrete approximations to local and global energy conservation laws for multi-symplectic PDEs with cubic invariants. ...
, and demonstrate their good stability properties and superior running speed when compared to fully implicit schemes. ...
The authors wish to express gratitude to Elena Celledoni and Brynjulf Owren for constructive discussions and helpful suggestions during our work on this paper, and to Benjamin Tapley for helping with the ...
arXiv:1907.02122v2
fatcat:kb2qjoya65dexffyw2bhlf3fqm
A general framework for deriving integral preserving numerical methods for PDEs
[article]
2011
arXiv
pre-print
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In particular, linearly implicit methods preserving a time discretised version of the invariant is developed for systems of partial differential equations with polynomial nonlinearities. ...
The framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order. ...
The plot shows that for a given global error the linearly implicit scheme is computationally cheaper than the fully implicit scheme. ...
arXiv:1009.3151v2
fatcat:pbwpksomu5fadev5ydu3vspo7u
Linearly Implicit Global Energy Preserving Reduced-order Models for Cubic Hamiltonian Systems
[article]
2023
arXiv
pre-print
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For this, we present a linearly implicit global energy-preserving method to construct reduced-order models. ...
This work discusses the model reduction problem for large-scale multi-symplectic PDEs with cubic invariants. ...
Notice that the linearly implicit global energy preserving (LIGEP) method is proposed for a general skew-symmetric differential matrix D x in [4] . ...
arXiv:2308.02625v1
fatcat:s2cylcnvj5d43ork2yll3sx3lm
Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation
2021
Discrete and continuous dynamical systems. Series B
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A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box ...
It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. ...
Kahan's method [22] proposed for quadratic ODEs is linearly implicit. Its general form was written down in [23] and it was applied to Hamiltonian ODEs with cubic Hamiltonian in [6, 7, 8, 9] . ...
doi:10.3934/dcdsb.2021139
fatcat:jzdlt2oleveyxj6qjt52yj7feu
Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Canonical Hamiltonian Systems
[article]
2021
arXiv
pre-print
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2021 pre-print arXiv:2107.12996v1 fatcat:pstjrn5c6bbdfg34tg4whnkx6u
This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of canonical Hamiltonian systems. ...
Our numerical results demonstrate Hamiltonian operator inference on a linear wave equation, the cubic nonlinear Schrödinger equation, and a nonpolynomial sine-Gordon equation. ...
Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. ...
arXiv:2107.12996v2
fatcat:eju3n2hpenafrdunzcax4r3vb4
A new symmetric linearly implicit exponential integrator preserving polynomial invariants or Lyapunov functions for conservative or dissipative systems
[article]
2021
arXiv
pre-print
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We present a new linearly implicit exponential integrator that preserves the polynomial first integrals or Lyapunov functions for the conservative and dissipative stiff equations, respectively. ...
The numerical simulations confirm the conservative properties of the proposed method and demonstrate its good behavior in superior running speed when compared with fully implicit schemes for long-time ...
Acknowledgement The author would like to thank Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Geometry, compatibility and structure preservation ...
arXiv:2104.12118v1
fatcat:sjwa4hxqbncr3jcuk3bmoi7rny
Structure Preserving Model Order Reduction of Shallow Water Equations
[article]
2020
arXiv
pre-print
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as a partial differential equation (PDE) with quadratic nonlinearity by the linearly implicit Kahan's method. ...
We show that both methods preserve numerically the invariants like energy, the Casimirs like the enstrophy, mass, and circulation over a long time. ...
Acknowledgments The authors thank for the constructive comments of the referees which helped much to improve the paper. This work was supported by 100/2000 Ph.D. ...
arXiv:1907.09406v2
fatcat:xhokpi5z6jgzrfkqq7iukodc3y
Wave energy self-trapping by self-focusing in large molecular structures: A damped stochastic discrete nonlinear Schrödinger equation model
2007
Physica D : Non-linear phenomena
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Numerical results are presented for the first time for the discrete models including the highly nonlinear damping term, and new numerical methods are introduced for this purpose. ...
Previous studies directed at the SNLS approximations have indicated that the self-focusing of wave energy to highly localized states can be inhibited by phase noise (modeling thermal effects) and can be ...
Acknowledgements The first author gives thanks to Peter Christiansen and Yuri Gaididei for numerous discussions, to the IMM at the Danish Technical University for supporting several visits, and to the ...
doi:10.1016/j.physd.2006.08.024
fatcat:27jgwla5unaj5b27klqj34vbdq
A Note on the Symplectic Integration of the Nonlinear Schrödinger Equation
2004
Journal of Computational Electronics
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The long-term stability of a numerical method and its conservation properties is an important feature since it assures that the underlying physics of the solution are respected and it ensures that the ...
In this paper we describe symplectic integrators for the nonlinear Schrödinger equation with arbitrary potentials and perform numerical experiments comparing different approaches and highlighting their ...
0, 1]
Theorem 2. 4 ( 4 Long Term Energy Conservation) If a symplectic numerical method of order m with step size h is applied to a Hamiltonian system with analytic H : D → R (where D ⊂ R 2d ) and the ...
doi:10.1023/b:jcel.0000029454.06133.f9
fatcat:m6zf7qbalndmle6s24g3i772na
Poisson integrators
[article]
2005
arXiv
pre-print
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Numerical integrators using generating functions, Hamiltonian splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems and Hamiltonian systems with a general Poisson structure. ...
An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. ...
The author acknowledges the support of the Swiss National Science Foundation and is grateful to Ernst Hairer and Gerhard Wanner for their hospitality during his stay at Université de Genève, in 2000. ...
arXiv:nlin/0505037v1
fatcat:pqd2lwyml5glpgzqolslphqpi4
Poisson integrators
2004
Mathematical and computer modelling
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Numerical integrators using generating functions, Hamiltonian splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems and Hamiltonian systems with a general Poisson structure. ...
An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. ...
The author acknowledges the support of the Swiss National Science Foundation and is grateful to Ernst Hairer and Gerhard Wanner for their hospitality during his stay at Université de Genève, in 2000. ...
doi:10.1016/j.mcm.2005.01.015
fatcat:dpy6p6f7nzef3mizztiygjyzmq
A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation
[article]
2020
arXiv
pre-print
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Comparing with the original exponential energy-preserving integrator which usually leads to a nonlinear algebraic system, our new method only involve a linear system with constant coefficient matrix. ...
Taking the nonlinear Klein-Gordon equation for example, we derive the concrete energy-preserving scheme and demonstrate its high efficiency through numerical experiments. ...
Acknowledgments The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions. ...
arXiv:1908.10265v2
fatcat:pbwoedr43jdkbb667e3gnli23y
Geometric integrators for ODEs
2006
Journal of Physics A: Mathematical and General
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In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. ...
Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. ...
Acknowledgments The authors are grateful to W-J Beyn and Shang Zai-jiu for useful correspondence, and to David McLaren, Stephen Marsland, Dion O'Neale, Brett Ryland, Will Wright, and Philip Zhang for their ...
doi:10.1088/0305-4470/39/19/s01
fatcat:2m2xrbmywjawldihzjxpdie4nm
On nonparaxial nonlinear Schrödinger-type equations
2019
Journal of Computational and Applied Mathematics
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In this sense, different numerical procedures that preserve the Hamiltonian and multi-symplectic structures are discussed and illustrated with numerical experiments. ...
First, some theoretical results on linear well-posedness, Hamiltonian and multi-symplectic formulations are derived. ...
Acknowledgments This work has been supported by Ministerio de Ciencia, Innovación y Universidades, FEDER and Junta de Castilla y León through projects MTM 2015-66837-P, VA024P17, VA041P17 and VA105G18. ...
doi:10.1016/j.cam.2019.02.029
fatcat:v4yks7puf5fz7kvcnd2euq4frq
On nonparaxial nonlinear Schrödinger-type equations
[article]
2019
arXiv
pre-print
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In this sense, different numerical procedures that preserve the Hamiltonian and multi-symplectic structures are discussed and illustrated with numerical experiments. ...
First, some theoretical results on linear well-posedness, Hamiltonian and multi-symplectic formulations are derived. ...
Acknowledgments This work has been supported by Ministerio de Ciencia, Innovacin y Universidades, FEDER and Junta de Castilla y León through projects MTM 2015-66837-P, VA024P17, VA041P17 and VA105G18. ...
arXiv:1902.08462v1
fatcat:hi4njxofpzbadfe4odxtc4hfzy
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