A Hamiltonian, explicit algorithm with spectral accuracy for the ‘good’ Boussinesq system

Frutos, Javier de; Ortega, Tomás; Sanz‐Serna, J. M.

doi:10.1016/0045-7825(90)90046-o

Cited by 19 publications

(9 citation statements)

References 11 publications

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“…This is the approach taken, for example, by Feng and Qin (1987) for the linear wave equation, and by de Frutos, Ortega, and Sanz-Serna (1990) for the Boussinesq equation. Instead of discretizing the PDE directly, we discretize both the Hamiltonian function and the Hamiltonian (Poisson) structure, then form the resulting ODE's.…”

Section: Introductionmentioning

confidence: 99%

Symplectic integration of Hamiltonian wave equations

1993

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The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which the best results can be obtained; in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.Mathematics Subject Classification (1991): 65M20, 58F05, 35L70

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Section: Introductionmentioning

confidence: 99%

Symplectic integration of Hamiltonian wave equations

1993

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“…Ismail and Bratsos [24] proposed a predictor-corrector (P-C) pair to solve the BS equation using a fourth order in time and second order in space scheme and El-Zoheiry [17] studied the GB equation using an iterative implicit finite-difference scheme. FEM were found in Manoranjan et al [29] and Pani and Saranga [35] who used the Galerkin method to study the GB equation, pseudo-spectral methods in Defrutos et al [15] and Defrutos et al [16] and the Adomian decomposition method in Wazwaz [42] and Attili [4]. Finally, an extended interest has been observed in the real world problems, as for example the study of the propagation of shallow sea waves, where the Boussinesq type equations are applied -see for instance Bratsos et al [13] and the references given there in.…”

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confidence: 97%

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

Bratsos

2007

Numer Algor

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A predictor-corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrixexponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.

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“…Hereafter, we shall assume that u 0 (x) and v 0 (x) are such that the solution of problem (4) and (5) is regular enough, as a periodic function on [a, b], for all t ≥ 0. The numerical solution of (1), (3) or (4) has been developed along different directions, ranging from the pseudo-spectral or splitting approach [13][14][15][16][17][18][19]46], up to finite-difference and finite-element schemes [20][21][22][23][24]47], as well as structure-preserving methods [10,25,26] and energy-preserving methods [27,28]. In particular, [11,12] consider an energy-conserving strategy based on the Hamiltonian boundary value methods (HBVMs) for the "good" Boussinesq and the improved Boussinesq equation, respectively, while a second-order symplectic method preserving the energy and the momentum is considered in [29].…”

Section: Introductionmentioning

confidence: 99%

Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

Brugnano

Gurioli

Zhang

2019

Numerical Methods Partial

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In this paper we study the geometric numerical solution of the so called "good" Boussinesq equation. This goal is achieved by using a convenient space semi-discretization, able to preserve the corresponding Hamiltonian structure, then using energy-conserving Runge-Kutta methods in the Hamiltonian boundary value method class for the time integration. Numerical tests are reported, confirming the effectiveness of the proposed method.

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A Hamiltonian, explicit algorithm with spectral accuracy for the ‘good’ Boussinesq system

Cited by 19 publications

References 11 publications

Symplectic integration of Hamiltonian wave equations

Symplectic integration of Hamiltonian wave equations

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

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