Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations

Zhao, Chengchao; Liu, Nan; Ma, Yong; Zhang, Jiwei

doi:10.4310/cms.2023.v21.n3.a7

Cited by 8 publications

(3 citation statements)

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“…On the one hand, since a first-order scheme is used in the first time step, a direct use of the standard discrete Grönwall inequality for the error estimate in the H 1 -norm would lead to order reduction. This order reduction of O(τ 1.5 ) has been observed for linear parabolic equations [27,31] and N-S equations [28]. Recently, Ma et.…”

Section: Introductionsupporting

confidence: 66%

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A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

Di¹,

Ma²,

Shen³

et al. 2023

ESAIM: M2AN

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We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxil- iary variable (SAV) scheme for Navier-Stokes equation with periodic boundary conditions [11, Huang and Shen, SIAM J. Numer. Anal., 2021] to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H 1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solv- ing Navier-Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and eﬀiciency of our proposed methods.

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Section: Introductionsupporting

confidence: 66%

“…al. studied the unconditionally optimal convergence in L 2 -norm for general semi-linear equations [31].…”

Section: Introductionmentioning

confidence: 99%

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

Di¹,

Ma²,

Shen³

et al. 2023

ESAIM: M2AN

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show abstract

“…Very recently, the authors of [19] proposed a linear second-order difference method for solving the Allen-Cahn equation, and they derived the 𝐿 ∞ error estimate of the proposed scheme under 𝑟 𝑘 < 1+ √ 2. In [52], Zhao et al presented a linearized variable-step BDF2 scheme for solving nonlinear parabolic equation, and they proved the unconditional error estimate under 𝑟 𝑘 < 4.8645 and the maximum temporal stepsize 𝜏 ≤ 𝐶 1 √ 𝑁 by adopting the error splitting approach. After then, Li et al extended this method to solve a nonlinear Ginzburg-Landau equation [44] and coupled Ginzburg-Landau equations [27] under the same conditions.…”

mentioning

confidence: 99%

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Zhang,

2024

ESAIM: M2AN

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Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies $ r_{k} := \tau_{k}/\tau_{k-1} < 4.8645 $. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction $ r_{k} < 4.8645 $ on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

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An Implicit–Explicit Second-Order BDF Numerical Scheme with Variable Steps for Gradient Flows

Hou

Qiao

2023

J Sci Comput

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In this paper, we propose and analyze an efficient implicit-explicit (IMEX) secondorder backward differentiation formulation (BDF2) scheme with variable time steps for gradient flow problems using a scalar auxiliary variable (SAV) approach. Comparing with the traditional second-order SAV approach [Shen et al., J. Comput. Phys. 2018], we only use a first-order method to approximate the auxiliary variable. This treatment does not affect the second-order accuracy of the unknown function ϕ, and is essentially important for deriving the unconditional energy stability of the proposed BDF2 scheme with variable time steps. We prove the unconditional energy stability of the scheme for a modified discrete energy with the adjacent time step ratio γn+1 := τn+1/τn ≤ 4.8645. The uniform H 2 bound for the numerical solution is derived under a mild regularity restriction on the initial condition, that is ϕ(x, 0) ∈ H 2 . Based on this uniform bound of the numerical solution, a rigorous error estimate of the proposed scheme is carried out on the nonuniform temporal mesh. Finally, serval numerical tests are provided to validate the theoretical claims. With the application of an adaptive time-stepping strategy, the efficiency of our proposed scheme can be clearly observed in the coarsening dynamics simulation.

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Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations

Cited by 8 publications

References 0 publications

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

An Implicit–Explicit Second-Order BDF Numerical Scheme with Variable Steps for Gradient Flows

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