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Linearly implicit structure-preserving schemes for Hamiltonian systems
[article]
2020
arXiv
pre-print
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Kahan's method and a two-step generalization of the discrete gradient method are both linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a cubic Hamiltonian. ...
The schemes are applied to the Korteweg-de Vries equation and the Camassa-Holm equation, and the numerical results are presented and analysed. ...
Acknowledgements The authors would like to thank Elena Celledoni, Takayasu Matsuo and Brynjulf Owren for initiating the work that led to this paper, and for their very helpful comments on the manuscript ...
arXiv:1901.03573v3
fatcat:6u7hesaixzbollstwbeljcwdb4
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. ...
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. ...
See Section 6 for another example that tests the long time structure preserving properties of these schemes. ...
arXiv:1009.3151v2
fatcat:pbwpksomu5fadev5ydu3vspo7u
High-order linearly implicit structure-preserving exponential integrators for the nonlinear Schrödinger equation
[article]
2021
arXiv
pre-print
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A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger equation. ...
Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other existing structure-preserving schemes. ...
Acknowledgments The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions. ...
arXiv:2103.00390v2
fatcat:7gbqrze5qffhpl43mwieu2l25y
A General Framework for Deriving Integral Preserving Numerical Methods for PDEs
2011
SIAM Journal on Scientific Computing
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In particular, linearly implicit methods preserving a time averaged version of the invariant is developed for systems of partial differential equations with polynomial nonlinearities. ...
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. ...
In Figure 1 .1 we plot the global error versus the number of linear solves for the two schemes. The linearly implicit scheme solves one linear system in each time step. ...
doi:10.1137/100810174
fatcat:r3gxu676jvbn3eurlu42k2izni
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. ...
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
Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions
[article]
2018
arXiv
pre-print
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Further combining the symplectic Runge-Kutta method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional ...
This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving ...
Therefore, it is preferable to construct linearly implicit schemes through the SAV approach for large scale simulations, keeping the system energy being preserved as well. ...
arXiv:1809.02704v1
fatcat:iv4uqhg7sbgb3fvvyvoqixao3a
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
Relaxation Runge-Kutta Methods for Hamiltonian Problems
[article]
2020
arXiv
pre-print
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We also prove that these methods are superconvergent for a certain class of Hamiltonian systems. ...
The recently-introduced relaxation approach for Runge-Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. ...
The authours would like to thank Ernst Hairer for a discussion of ...
arXiv:2001.04826v2
fatcat:scglmbuhqbcfvkzzak7zentbzi
Structure-preserving algorithms for multi-dimensional fractional Klein-Gordon-Schrödinger equation
[article]
2019
arXiv
pre-print
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This paper aims to construct structure-preserving numerical schemes for multi-dimensional space fractional Klein-Gordon-Schrödinger equation, which are based on the newly developed partitioned averaged ...
First, we derive an equivalent equation, and reformulate the equation as a canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. ...
The Hamiltonian structure is important to analyse the conservative systems and further to construct numerical schemes for them. ...
arXiv:1911.10845v2
fatcat:a5ffnywksza6jbzzgah37m57b4
Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation
2022
Fractal and Fractional
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After that, linearly implicit energy-preserving schemes which have high accuracy are given by applying the Runge–Kutta method to approximate the semi-discrete system in temporal direction and using the ...
In this paper, a family of high-order linearly implicit exponential integrators conservative schemes is constructed for solving the multi-dimensional nonlinear fractional Schrödinger equation. ...
The method can conserve some intrinsic properties of the dynamical system, which we called the structure-preserving algorithm, and scholars have developed structure-preserving schemes for system (1) . ...
doi:10.3390/fractalfract6050243
fatcat:zxgb4dxs3bf6xjb4noxp472kdi
A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation
[article]
2020
arXiv
pre-print
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Comput. 38(2016) A1876-A1895] for conservative systems, which now becomes linearly implicit by further utilizing the idea of the scalar auxiliary variable approach. ...
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
On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry
1996
Computer Methods in Applied Mechanics and Engineering
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This paper presents a detailed comparison of two implicit time integration schemes for a simple non-linear Hamiltonian system with symmetry: the motion of a particle in a central force field. ...
conserving scheme, and to compare the two schemes with respect to accuracy. ...
Introduction and motivation In this paper we present a detailed comparison of two implicit time-integration schemes for a simple non-linear Hamiltonian system with symmetry: the motion of a particle in ...
doi:10.1016/0045-7825(96)01009-2
fatcat:wgtybixdcreazgx2ivjwhw26qa
Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems
[article]
2021
arXiv
pre-print
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Specifically, one part of the derived schemes is totally explicit, and the other is linearly implicit. ...
In this paper, we develop a framework to construct energy-preserving methods for multi-components Hamiltonian systems, combining the exponential integrator and the partitioned averaged vector field method ...
To the best of our knowledge, the only linearly implicit EP-EI for Hamiltonian systems is constructed with the help of the energy quadratization approach and thus only preserves a modified energy [30] ...
arXiv:2110.04092v2
fatcat:ysyh3vim7famtauq74myzgub5m
An explicit and practically invariants-preserving method for conservative systems
[article]
2020
arXiv
pre-print
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An explicit numerical strategy that practically preserves invariants is derived for conservative systems by combining an explicit high-order Runge-Kutta (RK) scheme with a simple modification of the standard ...
the fully implicit integrators. ...
In contrast to fully implicit schemes, linearly implicit methods only involve to solve a linear system at each time step, and therefore have attracted continuous attention in recent years. ...
arXiv:2009.06877v1
fatcat:w5xra5aa6bfhrl443xeoawgcbq
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. ...
recast the KPI equation into a multisymplectic Hamiltonian formulation, and then develop a linearized implicit scheme preserving a discrete LECL for the equation. ...
doi:10.3934/dcdsb.2021139
fatcat:jzdlt2oleveyxj6qjt52yj7feu
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