Long-time Accurate Symmetrized Implicit-explicit BDF Methods for a Class of Parabolic Equations with Non-self-adjoint Operators

Li, Buyang; Wang, Kai

doi:10.1137/18m1227536

Cited by 14 publications

(10 citation statements)

References 32 publications

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“…For a proof by a spectral technique in the case of selfadjoint operators, we refer to [13, chapter 10]; for a proof in the general case, under a sharp condition on the nonselfadjointness of the operator as well as for nonlinear parabolic equations, by a combination of spectral and Fourier techniques, see, e.g., [2] and references therein. For a long-time estimate in the case of selfadjoint operators and an application to the Stokes-Darcy problem, see [10].…”

Section: Stabilitymentioning

confidence: 99%

The energy technique for the six-step BDF method

Akrivis

Chen

Fan

2020

Preprint

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In combination with the Grenander-Szegö theorem, we observe that a relaxed positivity condition on multipliers, milder than the basic requirement of the Nevanlinna-Odeh multipliers that the sum of the absolute values of their components is strictly less than 1, makes the energy technique applicable to the stability analysis of BDF methods for parabolic equations with selfadjoint elliptic part. This is particularly useful for the six-step BDF method for which no Nevanlinna-Odeh multiplier exists. We introduce multipliers satisfying the positivity property for the six-step BDF method and establish stability of the method for parabolic equations.

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Section: Stabilitymentioning

confidence: 99%

The energy technique for the six-step BDF method

Akrivis

Chen

Fan

2020

Preprint

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“…This is because the operator A = ∆ + f ′ (u 0 )I might not be negative definite, and hence the solution might blow up exponentially as T → ∞. In case that A is negative definite, we can obtain an error estimate which is uniform in large terminal time (e.g., [19,23]). Now we turn to the subdiffusion problem driven by a general source term:…”

Section: 2mentioning

confidence: 99%

High-order Time Stepping Schemes for Semilinear Subdiffusion Equations

Wang¹

2020

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The aim of this paper is to develop and analyze high-order time stepping schemes for solving semilinear subdiffusion equations. We apply the k-step BDF convolution quadrature to discretize the time-fractional derivative with order α ∈ (0, 1), and modify the starting steps in order to achieve optimal convergence rate. This method has already been well-studied for the linear fractional evolution equations in Jin, Li and Zhou [19], while the numerical analysis for the nonlinear problem is still missing in the literature. By splitting the nonlinear potential term into an irregular linear part and a smoother nonlinear part, and using the generating function technique, we prove that the convergence order of the corrected BDFk scheme is O(τ min(k,1+2α−ǫ) ), without imposing further assumption on the regularity of the solution. Numerical examples are provided to support our theoretical results.

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“…Let {t n = τ n} be a uniform partition of the interval [0, T ], with a time step size τ = T /N . For n ≥ 1, the k-step BDF scheme seeks U n ∈ V such that [30] ∂τ…”

mentioning

confidence: 99%

“…Then ∂τ u(t n ) is the standard approximation of ∂ t u(t n ) by BDFk. For 1 ≤ n ≤ k − 1, these constants have been determined in [24,30], cf. Table 2.1.…”

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confidence: 99%

A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations

Wu¹

2020

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In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations. In particular, we apply the k-step backward differentiation formula, and then develop an iterative solver by using the waveform relaxation technique. Each resulting iterate represents a periodic-like system, which could be further solved in parallel by using the diagonalization technique. The convergence of the waveform relaxation iteration is theoretically examined by using the generating function method. The approach we established in this paper extends the existing argument of single-step methods in Gander and Wu [Numer. Math., 143 (2019), pp. 489-527] to general BDF methods up to order 6. The argument could be further applied to the time-fractional subdiffusion equation, whose discretization shares common properties of the standard BDF methods, because of the nonlocality of the fractional differential operator. Illustrative numerical results are presented to complement the theoretical analysis.

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Long-time Accurate Symmetrized Implicit-explicit BDF Methods for a Class of Parabolic Equations with Non-self-adjoint Operators

Cited by 14 publications

References 32 publications

The energy technique for the six-step BDF method

The energy technique for the six-step BDF method

High-order Time Stepping Schemes for Semilinear Subdiffusion Equations

A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations

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