A <scp>Fourier</scp> pseudospectral method for the “good” <scp>Boussinesq</scp> equation with second‐order temporal accuracy

Cheng, Kelong; Feng, Wenqiang; Gottlieb, Sigal; Wang, Cheng

doi:10.1002/num.21899

Cited by 70 publications

(37 citation statements)

References 60 publications

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“…But for models whose exact solutions hardly are found, the numerical method is an alternative choice. Especially, high-accuracy numerical algorithms [2,3,4,12,25], which maintain the conservative properties of the initial equation, could guarantee the validity of the numerical approximation. Because the high accuracy could guarantee the precision of the approximation, and the conservatives make the approximate solution reflect the physical phenomena better.…”

Section: Introductionmentioning

confidence: 99%

A high-accuracy conservative difference approximation for Rosenau-KdV equation

Hu¹,

Jun²,

Zhuo³

2017

J. Nonlinear Sci. Appl.

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In this paper, we study the initial-boundary value problem of Rosenau-KdV equation. A conservative two level nonlinear Crank-Nicolson difference scheme, which has the theoretical accuracy O(τ 2 + h 4 ), is proposed. The scheme simulates two conservative properties of the initial boundary value problem. Existence, uniqueness, and priori estimates of difference solution are obtained. Furthermore, we analyze the convergence and unconditional stability of the scheme by the energy method. Numerical experiments demonstrate the theoretical results.

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Section: Introductionmentioning

confidence: 99%

A high-accuracy conservative difference approximation for Rosenau-KdV equation

Hu¹,

Jun²,

Zhuo³

2017

J. Nonlinear Sci. Appl.

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“…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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“…We observe that the conservation of the linear invariants (6) and (7) is implicitly gained, by virtue of Theorem 4, through the definition of the semi-discrete problem obtained by truncating the series in (15) and (16). As matter of fact, their conservation is satisfied by all the above methods and, therefore, it will not be checked further.…”

Section: Numerical Testsmentioning

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%

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 Fourier pseudospectral method for the “good” Boussinesq equation with second‐order temporal accuracy

Abstract: In this article, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second-order time-stepping for the numerical solution of the "good" Boussinesq equation

Cited by 70 publications

References 60 publications

A high-accuracy conservative difference approximation for Rosenau-KdV equation

A high-accuracy conservative difference approximation for Rosenau-KdV equation

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

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

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