Optimal a posteriori estimators for the variable step-size BDF2 method for linear parabolic equations

Wang, Wansheng; Mao, Mengli; Huang, Yi

doi:10.1016/j.cam.2022.114306

Cited by 12 publications

(1 citation statement)

References 23 publications

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“…A simple universal error estimator was developed for the GS4-1 family of HFF 33,12 algorithms (including the BDF2 scheme as a special case), which is also specifically designed for second-order accurate schemes (Wang et al, 2022b;Wang et al, 2021a). Most recently, an a posteriori error estimator based on BDF2 reconstruction of the piecewise linear approximate solutions was derived and the starting step size in the BDF2 was investigated (Wang et al, 2022a). Toward a unified design on the error estimator for BDF methods from low to high order, Hay et al (2015aHay et al ( , 2015b proposed an accurate error estimator for BDF methods with hprefinement merits in the time dimension.…”

Section: Introductionmentioning

confidence: 99%

Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping

Wang,

Luo,

Zhang

et al. 2023

HFF

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Purpose The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer. Design/methodology/approach The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms. Findings The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations. Originality/value In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.

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Section: Introductionmentioning

confidence: 99%

Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping

Wang,

Luo,

Zhang

et al. 2023

HFF

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A Posteriori Error Estimates for Exponential Midpoint Integrator Finite Element Method for Parabolic Equations

Hu,

Wang,

Fang

2024

Math Methods in App Sciences

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In this paper, a posteriori error estimates for the fully discrete approximation for linear and semilinear parabolic equations are established. The exponential midpoint method is used for the temporal discretization, and the spatial discretization takes the standard linear elements. By introducing a continuous piecewise quadratic reconstruction, the error of the temporal discretization is derived. The error of the spatial discretization is evaluated by the interpolation estimates. Using the energy technique, we derive residual‐based a posteriori lower and upper bounds for the fully discrete approximation. Our theoretical results are supported by extensive numerical experiments.

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Robust Error Estimates for Second‐Order Stabilization Finite Element Method for Navier–Stokes Equations With Small Viscosity and Nonsmooth Initial Data

Yang,

Wang,

Jin

2024

Numerical Methods Partial

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In this article, we study the fully discrete finite element approximation to the two‐dimensional Navier–Stokes equations with a small viscosity coefficient and initial data by using semi‐implicit two‐step backward differentiation formulae (BDF2) method with variable step‐sizes and grad‐div stabilization. The variable step sizes allow us to take special viscosity‐dependent, locally refined time step sizes and thus prove that the variable step‐size BDF2 method can achieve second‐order convergence in time. By adding the grad‐div stabilization, we obtain a robust, fully discrete error bound in which the constants reduce dependence on inverse powers of the viscosity. Numerical results illustrate the effectiveness of the proposed method for such types of Navier–Stokes equations.

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Optimal a posteriori estimators for the variable step-size BDF2 method for linear parabolic equations

Cited by 12 publications

References 23 publications

Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping

Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping

A Posteriori Error Estimates for Exponential Midpoint Integrator Finite Element Method for Parabolic Equations

Robust Error Estimates for Second‐Order Stabilization Finite Element Method for Navier–Stokes Equations With Small Viscosity and Nonsmooth Initial Data

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