Local structure-preserving algorithms for partial differential equations

Wang, Yushun; Wang, Bin; Qin, Meng

doi:10.1007/s11425-008-0046-7

Cited by 45 publications

(29 citation statements)

References 22 publications

(20 reference statements)

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“…The other approach is based on a reformulation of the Hamiltonian PDE into a multi-symplectic form, which provides the PDE with three local conservation laws: the multi-symplectic conservation law, the energy conservation and the momentum conservation law [10,11,12]. Then one may consider methods that preserve the local conservation laws, see for example [13]. These locally defined properties are not dependent on the choice of boundary conditions, giving the methods that preserve local energy an advantage over methods that preserve a global energy, especially since local conservation laws will always lead to global conservation laws whenever periodic boundary conditions are considered.…”

Section: Introductionmentioning

confidence: 99%

Linearly implicit structure-preserving schemes for Hamiltonian systems

Eidnes¹,

Li²,

Sato

2021

Journal of Computational and Applied Mathematics

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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. The methods are tested on the one-dimensional Korteweg-de Vries equation and the two-dimensional Zakharov-Kuznetsov equation; the numerical simulations confirm the conservative properties of the methods, and demonstrate their good stability properties and superior running speed when compared to fully implicit schemes.

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Section: Introductionmentioning

confidence: 99%

Linearly implicit structure-preserving schemes for Hamiltonian systems

Eidnes¹,

Li²,

Sato

2021

Journal of Computational and Applied Mathematics

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“…We can get multi-symplectic algorithms, local energy-preserving algorithms, and local momentum-preserving algorithms in this way. 37 The other one is called method of lines, which first discrete the PDE in spatial direction to yield a semi-discrete Hamiltonian ODEs and then integrate the resulted ODEs by some proper structure-preserving algorithm. Nevertheless, no method is mature to ensure the generated ODEs to be Hamiltonian, especially the canonical Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

An SDG Galerkin structure‐preserving scheme for the Klein‐Gordon‐Schrödinger equation

Wang

2020

Math Methods in App Sciences

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In this paper, we use the Galerkin weak form to construct a structure-preserving scheme for Klein-Gordon-Schrödinger equation and analyze its conservative and convergent properties. We first discretize the underlying equation in space direction via a selected finite element method, and the Hamiltonian partial differential equation can be casted into Hamiltonian ordinary differential equations based on the weak form of the system afterwards. Then, the resulted ordinary differential equations are solved by the symmetric discrete gradient method, which yields a charge-preserving and energy-preserving scheme. Moreover, the numerical solution of the proposed scheme is proved to be bounded in the discrete L ∞ norm and convergent with the convergence order of (h 2 + 2 ) in the discrete L 2 norm without any grid ratio restrictions, where h and are space and time step, respectively. Numerical experiments conducted last to verify the theoretical analysis. KEYWORDS finite element method, Klein-Gordon-Schrödinger equation, structure-preserving algorithm, symmetric discrete gradient method MSC CLASSIFICATION 37K05; 37K40; 65M12; 65M60; 74S05 Math Meth Appl Sci. 2020;43:6011-6030. wileyonlinelibrary.com/journal/mmaAdding Equations (33) and (36) yields Equation (28), ie, the energy conservation law.Theorem 3.4. The solution of the proposed SDG Galerkin structure-preserving scheme (Equations 23-26) satisfiesProof. According to Equations (27) to (28), we have). Then we have the L 2 error estimates as follows:Proof. According to the Equations (24) and (26), we eliminate the intermediate variable v and getwhereNote that we consider the case that the exact solution is bounded in the discrete L ∞ norm, ie, ||p n || ∞ ≤ C, ||q n || ∞ ≤ C, ||u n || ∞ ≤ C. According to Theorem 3.4, the boundedness of the numerical solution has been obtained, ie,

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“…By introducing some variables, the RLW‐type equation can be recast as a first‐order form of

true{\begin{array}{l} left & \frac{1}{2} u_{t} + \frac{a}{2} u_{x} + p_{x} = 0, \\ left & u_{t} = w, u_{x} = v, φ_{x} = u, \\ left & \frac{1}{2} φ_{t} - \frac{δ}{2} v_{t} + \frac{a}{2} φ_{x} - \frac{δ}{2} w_{x} = p - F' (u), \end{array}

where

F_{t} (u) / u_{t}

can be used instead of

F' (u)

. Simple derivations yield a local energy conservation law (LECL)

{true(F (u) + a u^{2} / 2 true)}_{t} - {(a φ_{t} u / 2 + φ_{t} p + δ w u_{t} / 2)}_{x} = 0.

As the LECL is independent of boundary conditions, it is more essential than the global one (appropriate boundary conditions result in the global ECL

I_{3}

). Naturally, we want to know whether there exists a scheme preserving the LECL in the discrete sense.…”

Section: Introductionmentioning

confidence: 99%

“…As the LECL is independent of boundary conditions, it is more essential than the global one [21] (appropriate boundary conditions result in the global ECL I 3 ). Naturally, we want to know whether there exists a scheme preserving the LECL in the discrete sense.…”

Section: Introductionmentioning

confidence: 99%

Efficient local structure‐preserving schemes for the RLW‐type equation

Cai

Hong

2017

Numerical Methods Partial

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The multisymplectic schemes have been used in numerical simulations for the RLW‐type equation successfully. They well preserve the local geometric property, but not other local conservation laws. In this article, we propose three novel efficient local structure‐preserving schemes for the RLW‐type equation, which preserve the local energy exactly on any time‐space region and can produce richer information of the original problem. The schemes will be mass‐ and energy‐preserving as the equation is imposed on appropriate boundary conditions. Numerical experiments are presented to verify the efficiency and invariant‐preserving property of the schemes. Comparisons with the existing nonconservative schemes are made to show the behavior of the energy affects the behavior of the solution.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1678–1691, 2017

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Local structure-preserving algorithms for partial differential equations

Cited by 45 publications

References 22 publications

Linearly implicit structure-preserving schemes for Hamiltonian systems

Linearly implicit structure-preserving schemes for Hamiltonian systems

An SDG Galerkin structure‐preserving scheme for the Klein‐Gordon‐Schrödinger equation

Efficient local structure‐preserving schemes for the RLW‐type equation

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