The time viscosity-splitting method for the Boussinesq problem

This paper presents decoupled second‐order accurate algorithm based on Crank–Nicolson LeapFrog (CNLF) scheme for the evolution Boussinesq equations. The proposed algorithm deals with the spatial discretization by finite element method and treat the temporal discretization by CNLF method. For the nonlinear term in the Boussinesq equations, the semi‐implicit method is used. Unconditional stability and error estimate of the numerical algorithm are proven. Some numerical tests are presented to justify the theoretical analysis.

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“…For the nonlinear terms in (), the skew‐symmetric trilinear forms [4, 8] are defined:

\forall \boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in X,\theta, \psi \in W

\begin{array}{cc}\hfill N\left(\boldsymbol{u};\boldsymbol{w},\boldsymbol{v}\right):&#x0003D; &amp; \kern0.2em \left(\left(\boldsymbol{u}\cdotp \nabla \right)\boldsymbol{w},\boldsymbol{v}\right)&#x0003D;\frac{1}{2}\left(\boldsymbol{u}\cdotp \nabla \boldsymbol{w},\boldsymbol{v}\right)-\frac{1}{2}\left(\boldsymbol{u}\cdotp \nabla \boldsymbol{v},\boldsymbol{w}\right).\hfill \\ {}\hfill \overline{N}\left(\boldsymbol{u};\theta, \psi \right):&#x0003D; &amp; \kern0.2em \left(\left(\boldsymbol{u}\cdotp \nabla \right)\theta, \psi \right)&#x0003D;\frac{1}{2}\left(\boldsymbol{u}\cdotp \nabla \theta, \psi \right)-\frac{1}{2}\left(\boldsymbol{u}\cdotp \nabla \psi, \theta \right).\hfill \end{array}

The following equalities and inequalities can be obtained from [7, 22, 23]:

\begin{matrix} right N (u; v, \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

Decoupled Crank–Nicolson LeapFrog method for the evolution Boussinesq equations

Jia

Feng

2023

Math Methods in App Sciences

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“…Recently, more and more attention has been attracted for the projection methods of the time-dependent natural convection problem, we can refer to [34,35] for more details. However, these methods are mainly some semi-discrete projection methods for the system (1).…”

Section: Introductionmentioning

confidence: 99%

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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“…Due to the coupling of the velocity, pressure and the temperature among the evolution Boussinesq equations, finding the accurate numerical solution becomes a more difficult task. There were many literatures to study this problem in [4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…are corresponding error norms.Let the domain Ω = [0, 1] × [0, 1], the parameters ν = λ = k = 1.0, T = 1.0 and true solutions in[7] u, p, θ are: u = (u1, u2) = (10x 2 (x − 1) 2 y(y − 1)(2y − 1)cos(t), −10y 2 (y − 1) 2 x(x − 1)(2x − 1)cos(t)), p = 10(2x − 1)(2y − 1)cos(t), θ = 10x 2 (x − 1) 2 y(y − 1)(2y − 1)cos(t) − 10y 2 (y − 1) 2 x(x − 1)(2x − 1)cos(t).…”

mentioning

confidence: 99%

Decoupled Crank-Nicolson LeapFrog Methods for the Evolution Boussinesq equations

Jia

Hong

2022

Preprint

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This paper presents decoupled second-order accurate algorithms based on Crank-Nicolson LeapFrog (CNLF) scheme for the evolution Boussinesq equations. The proposed algorithms deal with the spatial discretization by finite element method and treat the temporal discretization by CNLF method. For the nonlinear term in the Boussinesq equations, the semi-implicit method is used. Unconditional stability and error estimate of the numerical algorithm are proven. Some numerical tests are presented to justify the theoretical analysis.

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The time viscosity-splitting method for the Boussinesq problem

Cited by 5 publications

References 33 publications

Decoupled Crank–Nicolson LeapFrog method for the evolution Boussinesq equations

Decoupled Crank–Nicolson LeapFrog method for the evolution Boussinesq equations

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

Decoupled Crank-Nicolson LeapFrog Methods for the Evolution Boussinesq equations

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