Long Time Stability of High Order MultiStep Numerical Schemes for Two-Dimensional Incompressible Navier--Stokes Equations

Cheng, Kelong; Wang, Cheng

doi:10.1137/16m1061588

Cited by 34 publications

(13 citation statements)

References 41 publications

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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 [2,12,14,16,28,29,34,65,66], etc.…”

Section: Review Of Fourier Pseudo-spectral Approximationsmentioning

confidence: 99%

Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation

sci¹

2021

AAMM

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A second order accurate (in time) numerical scheme is analyzed for the slope-selection (SS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a modified Crank-Nicolson is applied for the surface diffusion part. The energy stability could be derived a modified form, in comparison with the standard Crank-Nicolson approximation to the surface diffusion term. Such an energy stability leads to an H 2 bound for the numerical solution. In addition, this H 2 bound is not sufficient for the optimal rate convergence analysis, and we establish a uniform-in-time H 3 bound for the numerical solution, based on the higher order Sobolev norm estimate, combined with repeated applications of discrete H ölder inequality and nonlinear embeddings in the Fourier pseudo-spectral space. This discrete H 3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.

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Section: Review Of Fourier Pseudo-spectral Approximationsmentioning

confidence: 99%

Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation

sci¹

2021

AAMM

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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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“…The proof can be found in [22]; also see the related analysis works in [8,9,11,12,13,14,21,25,35,36], etc.…”

mentioning

confidence: 94%

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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Long Time Stability of High Order MultiStep Numerical Schemes for Two-Dimensional Incompressible Navier--Stokes Equations

Cited by 34 publications

References 41 publications

Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation

Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film 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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