Line Integral Methods for Conservative Problems

Brugnano, Luigi; Iavernaro, Felice

doi:10.1201/b19319

Cited by 122 publications

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“…The next result summarizes the properties of HBVMs sketched above, where we also take into account that the abscissae are symmetrically distributed in the interval [0, 1] (we refer to [1,18] for full details).…”

Section: Hamiltonian Boundary Value Methodsmentioning

confidence: 93%

“…We here sketch the main facts concerning the so called blended implementation of HBVMs, a Newton-like iteration alternative to (36), which only requires to factor a matrix having the same size as that of the continuous problem, thus resulting into a much more efficient implementation of the methods [1,16]. This technique derives from the definition of blended implicit methods, which have been at first considered in [41,42], and then developed in [43][44][45].…”

Section: Blended Implementation Of Hbvmsmentioning

confidence: 99%

“…In many relevant instances, the solution has important geometric properties and the name geometric integrator has been coined to denote a numerical method able to preserve them (see, e.g., the monographs [1][2][3][4]). Often, the geometric properties of the vector field are summarized by the presence of constants of motion, namely functions of the state vector which are conserved along the solution trajectory of (1). For this reason, in such a case one speaks about a conservative problem.…”

Section: Introductionmentioning

confidence: 99%

“…along the solution y(t) of (1). Hereafter, we shall assume that both f : D → R m and C : D → R are suitably smooth (e.g., analytical).…”

Section: Introductionmentioning

confidence: 99%

“…Further generalizations, along several directions, have been considered in [19][20][21][22][23][24][25][26][27][28]: in particular, in [19] Hamiltonian boundary value problems have been considered, which are not covered in this review. The main reference on line integral methods is given by the monograph [1].…”

Section: Introductionmentioning

confidence: 99%

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Line Integral Solution of Differential Problems

Brugnano

Iavernaro

2018

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Abstract:In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references.

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Section: Hamiltonian Boundary Value Methodsmentioning

confidence: 93%

Section: Blended Implementation Of Hbvmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…along the solution y(t) of (1). Hereafter, we shall assume that both f : D → R m and C : D → R are suitably smooth (e.g., analytical).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Line Integral Solution of Differential Problems

Brugnano

Iavernaro

2018

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An energy-preserving Crank-Nicolson Galerkin method for Hamiltonian partial differential equations

Wang

Sheng

2016

Numer. Methods Partial Differential Eq.

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A semidiscretization based method for solving Hamiltonian partial differential equations is proposed in this article. Our key idea consists of two approaches. First, the underlying equation is discretized in space via a selected finite element method and the Hamiltonian PDE can thus be casted to Hamiltonian ODEs based on the weak formulation of the system. Second, the resulting ordinary differential system is solved by an energy-preserving integrator. The relay leads to a fully discretized and energy-preserved scheme. This strategy is fully realized for solving a nonlinear Schrödinger equation through a combination of the Galerkin discretization in space and a Crank-Nicolson scheme in time. The order of convergence of our new method is O(τ 2 + h 2 ) if the discrete L 2 -norm is employed. An error estimate is acquired and analyzed without grid ratio restrictions. Numerical examples are given to further illustrate the conservation and convergence of the energy-preserving scheme constructed.

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High‐order energy‐preserving schemes for the improved Boussinesq equation

Yan

Zhang

Zhao

et al. 2018

Numerical Methods Partial

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This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.

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Line Integral Methods for Conservative Problems

Cited by 122 publications

References 0 publications

Line Integral Solution of Differential Problems

Line Integral Solution of Differential Problems

An energy-preserving Crank-Nicolson Galerkin method for Hamiltonian partial differential equations

High‐order energy‐preserving schemes for the improved Boussinesq equation

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