On error estimates of the projection method for the time-dependent natural convection problem: First order scheme

Qian, Yanxia; Zhang, Tong

doi:10.1016/j.camwa.2016.07.013

Cited by 16 publications

(8 citation statements)

References 30 publications

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“…We have proved the convergence for R n u1 and R n θ in [12]. For the term R n u2 , we can easily find…”

Section: Error Estimates Of the Projection Schemesmentioning

confidence: 71%

“…In order to overcome this difficulty, Zhang and his co-authors considered the decoupled schemes for the natural convection problem in [22], [23], [25], and some meaningful results have been established. Recently, Qian and Zhang in [11], [12] considered the first order and higher order projection schemes for the time-dependent natural convection problem.…”

Section: Introductionmentioning

confidence: 99%

“…By the projection schemes, problem (1.1) is decoupled into two small linear subproblems, and each subproblem is solved more easily than the original one. For instance, the following modified projection schemes are analyzed in [11]:…”

Section: Introductionmentioning

confidence: 99%

“…We can readily check that (1.3) is equivalent to u n+1 = P H u n+1 , which explains why we call (1.2)-(1.3) the projection schemes. In [11], we developed the classical projection schemes and the modified projection schemes for the time-dependent natural convection problem (1.1). For the classical schemes, we established the convergence of weakly first order for the velocity and temperature and of weakly order 1 2 for the pressure.…”

Section: Introductionmentioning

confidence: 99%

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The second order projection method in time for the time-dependent natural convection problem

Qian

Zhang

2016

Appl Math

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“…We have proved the convergence for R n u1 and R n θ in [12]. For the term R n u2 , we can easily find…”

Section: Error Estimates Of the Projection Schemesmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The second order projection method in time for the time-dependent natural convection problem

Qian

Zhang

2016

Appl Math

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“…Among these numerical schemes, the Crank‐Nicolson extrapolation scheme is almost unconditional stability, while Crank‐Nicolson/Adams‐Bashforth scheme requires some restrictions on the time step and mesh size. Therefore, in this paper, we consider the Crank‐Nicolson extrapolation scheme for the natural convection problem, our work is extension and supplement the previous works and provide some new stability and convergence results for the numerical solutions. At the same time, based on He, He and Li, and He et al, for the 3D Navier‐Stokes equations, we also consider the Crank‐Nicolson extrapolation scheme for 3D time‐dependent natural convection problem.…”

Section: Introductionmentioning

confidence: 97%

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

Liang

Zhang

2019

Math Methods in App Sciences

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In this paper, we consider the Crank-Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank-Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two-grid Crank-Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two-grid Crank-Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H 1 -norm and for pressure in L 2 -norm. However, the two-grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes. KEYWORDSCrank-Nicolson extrapolation scheme, error estimates, stability, the natural convection equations, two-grid method MSC CLASSIFICATION 65N15; 65N30; 76D07Math Meth Appl Sci. 2019;42:6165-6191.wileyonlinelibrary.com/journal/mma

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Error estimates of an operator‐splitting finite element method for the time‐dependent natural convection problem

Yang

Huang

Jiang

2022

Numerical Methods Partial

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In this article, an operator‐splitting (or a fractional‐step) finite element method is proposed for the numerical solution of the time‐dependent natural convection problem. We first analyze the time‐discrete fractional‐step scheme, which decouples the considered system into three subproblems, and each subproblem can be solved more easily than the original one. Under some mild regularity assumptions, the stability and error estimates for the velocity and the temperature are established rigorously. In addition, a full discrete fractional‐step scheme based on the time‐discrete scheme is proposed and the rigorous error analysis is presented. We derive the temporal–spatial error estimates of scriptO(normalΔt+h)$$ \mathcal{O}\left(\Delta t+h\right) $$ for the velocity and the temperature in the discrete space l2()H1∩l∞()L2$$ {l}^2\left({H}^1\right)\cap {l}^{\infty}\left({L}^2\right) $$ under the constraint Δt ≥ Ch. Numerical experiments are performed to confirm the theoretical predictions and demonstrate the efficiency of the method.

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On error estimates of the projection method for the time-dependent natural convection problem: First order scheme

Cited by 16 publications

References 30 publications

The second order projection method in time for the time-dependent natural convection problem

The second order projection method in time for the time-dependent natural convection problem

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

Error estimates of an operator‐splitting finite element method for the time‐dependent natural convection problem

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