On Energy Stable, Maximum-Principle Preserving, Second-Order BDF Scheme with Variable Steps for the Allen--Cahn Equation

Liao, Hong-lin; Tang, Tao; Zhou, Tao

doi:10.1137/19m1289157

Cited by 91 publications

(39 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 4.3. Besides the Crank-Nicolson approach employed here, there have been a few recent works of the second order BDF schemes for certain gradient flow models, such as polynomial version of Cahn-Hilliard [14,15,48,64], epitaxial thin film equation [32,39,46,51], square phase field crystal [16], in which the energy stability was theoretically established. A successful extension to the Cahn-Hilliard equation with Flory-Huggins energy has also been reported in [13,27].…”

Section: Remark 41mentioning

confidence: 99%

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

Chen¹

2022

CICP

View full text Add to dashboard Cite

In this paper we propose and analyze a second order accurate numerical scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme, which ensures the positivity-preserving property, i.e., the numerical value of the phase variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized stability analysis. A few numerical results, including both the constant-mobility and solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.

show abstract

Section: Remark 41mentioning

confidence: 99%

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

Chen¹

2022

CICP

View full text Add to dashboard Cite

show abstract

“…As an exception, in our previous work [18], we have presented a novel analysis for the nonuniform BDF2 scheme of the Allen-Cahn equation under the same condition r k < 1 + √ 2. In a very recent work [19], for the linear diffusion problem, the L 2 norm stability and convergence estimates are presented under a much improved zero-stability condition…”

Section: Introductionmentioning

confidence: 97%

Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

et al. 2021

Self Cite

View full text Add to dashboard Cite

In this work, we are concerned with the stability and convergence analysis of the second-order backward difference formula (BDF2) with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r k := τ k /τ k−1 < 3.561. Moreover, with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms, the L 2 norm stability and rigorous error estimates are established, under the same step-ratio constraint that ensures the energy stability, i.e., 0 < r k < 3.561. This is known to be the best result in the literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

show abstract

“…Another method is to employ high-order schemes in time to have the same accuracy with a relatively large time-step, for example, the Runge-Kutta methods [4], Crank-Nicolson method [9,27] and two-step backward differentiation formula (BDF2) [1,12,32]. Specifically, the BDF2 method has received much attention due to its strong stability (A-stable) [7,16,19,20,28,32,34,35].…”

Section: Introductionmentioning

confidence: 99%

Sharp error estimate of variable time-step IMEX BDF2 scheme for parabolic integro-differential equations with initial singularity arising in finance

Zhao¹,

Yang²,

Di³

et al. 2022

Preprint

View full text Add to dashboard Cite

The recently developed technique of DOC kernels has been a great success in the stability and convergence analysis for BDF2 scheme with variable time steps. However, such an analysis technique seems not directly applicable to problems with initial singularity. In the numerical simulations of solutions with initial singularity, variable time-steps schemes like the graded mesh are always adopted to achieve the optimal convergence, whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied. In this paper, we revisit the variable time-step implicit-explicit two-step backward differentiation formula (IMEX BDF2) scheme presented in [W. Wang, Y. Chen and H. Fang, SIAM J. Numer. Anal., 57 (2019), pp. 1289-1317] to compute the partial integro-differential equations (PIDEs) with initial singularity. We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r k := τ k /τ k−1 (k ≥ 3) < rmax = 4.8645 and a much mild requirement on the first ratio, i.e., r2 > 0. This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy, i.e., the graded mesh t k = T (k/N ) γ . In this situation, the convergence of order O(N − min{2,γα} ) is achieved with N and α respectively representing the total mesh points and indicating the regularity of the exact solution. This is, the optical convergence will be achieved by taking γopt = 2/α. Numerical examples are provided to demonstrate our theoretical analysis.

show abstract

On Energy Stable, Maximum-Principle Preserving, Second-Order BDF Scheme with Variable Steps for the Allen--Cahn Equation

Cited by 91 publications

References 46 publications

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

Sharp error estimate of variable time-step IMEX BDF2 scheme for parabolic integro-differential equations with initial singularity arising in finance

Contact Info

Product

Resources

About