A second order operator splitting numerical scheme for the “good” Boussinesq equation

Zhang, Cheng; Wang, Hui; Huang, Jingfang; Cheng, Wang; Yue, Xingye

doi:10.1016/j.apnum.2017.04.006

Cited by 63 publications

(24 citation statements)

References 55 publications

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“…which will be assumed hereafter. Consequently, (19) continues formally to hold, as well as the result of Theorem 4, even though now the truncated approximations to u and v do not satisfy Equations (4) anymore. However, in the spirit of Galerkin methods, by imposing the residual be orthogonal to the functional space spanned by the entries of (x), also the results of Theorems 5 and 6 continue formally to hold, with the only difference that now the truncated versions of H(q, p) and M(q, p) do not coincide, up to a constant, with the functionals (10) and (11), respectively.…”

Section: Theoremmentioning

confidence: 90%

“…This latter, in turn, is equivalent, up to a constant, to the functional (10), via the transformations (15)- (19).…”

Section: Theorem 5 System (4) Can Be Cast In Hamiltonian Form Asmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

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.

show abstract

Section: Theoremmentioning

confidence: 90%

“…This latter, in turn, is equivalent, up to a constant, to the functional (10), via the transformations (15)- (19).…”

Section: Theorem 5 System (4) Can Be Cast In Hamiltonian Form Asmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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show abstract

“…It is closely related to the Fourier spectral method, but complements the basis by an additional pseudo-spectral basis, which allows to represent functions on a quadrature grid. This simplifies the evaluation of certain operators, and can considerably speed up the calculation when using fast algorithms such as the fast Fourier transform (FFT); see the related descriptions in [5,10,13,14,29,30,35,55,56].…”

Section: The Numerical Scheme 21 Fourier Pseudo-spectral Approximationsmentioning

confidence: 99%

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

Cheng¹,

Wang²,

Wise³

2019

CICP

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In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure is respected, and summation-by-parts formulae enable us to study the discrete energy stability for such a high-order spatial discretization. In the temporal approximation, a second order BDF stencil is applied, combined with an appropriate extrapolation for the concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to the artificial linear diffusion coming at an order lower that that of surface diffusion term. Such a choice leads to reduced numerical dissipation. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived in the ∞ (0, T ; 2 ) ∩ 2 (0, T ; H 3 N ) norm. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

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“…A more careful analysis reveals a key difficulty to obtain a higher order error estimate in the energy norm: lack of aliasing error control tool to estimate the numerical error term associated with D 2 N (u 2 ). In this paper, we make use of an aliasing error control technique for the Fourier pseudo-spectral method, developed in [22] and extensively applied in other related works [8,9,14,21,36], so that a convergence estimate in a higher order energy norm in derived: the H 2 error estimate for the variable u, combined with the discrete 2 error estimate for u t , in the full O(∆t 2 + h m ) accuracy order. This is a great improvement over the result reported in [19], in addition to the unconditional convergence estimate.…”

mentioning

confidence: 99%

A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation

Xia¹,

Yang²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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The nonlinear stability and convergence of a numerical scheme for the "Good" Boussinesq equation is provided in this article, with second order temporal accuracy and Fourier pseudo-spectral approximation in space. Instead of introducing an intermediate variable ψ to approximate the first order temporal derivative, we apply a direct approximation to the second order temporal derivative, which in turn leads to a reduction of the intermediate numerical variable and improvement in computational efficiency. A careful analysis reveals an unconditional stability and convergence for such a temporal discretization. In addition, by making use of the techniques of aliasing error control, we obtain an ∞ (0, T * ; H 2 ) convergence for u and ∞ (0, T * ; 2 ) convergence for the discrete time-derivative of the solution in this paper, in comparison with the ∞ (0, T * ; 2 ) convergence for u and the ∞ (0, T * ; H −2 ) convergence for the time-derivative, given in [19].

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A second order operator splitting numerical scheme for the “good” Boussinesq equation

Cited by 63 publications

References 55 publications

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

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

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation

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