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非定常渗透对流模型一阶分数步长算法的时间误差估计
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Abstract:
本文研究了求解非定常渗透对流模型的一阶分数步长时间离散算法，该方程是由非定常不可压缩Navier-Stokes方程和热传导方程所耦合的非线性多物理场模型。该算法的优点在于将Navier-Stokes方程的非线性性和不可压缩性进行分离，实现算法的高效性。理论上，在解的正则性假设下，我们得到了速度场和温度场一阶时间收敛阶。最后通过数值算例验证了所得到的收敛性结果。
In this paper, the first-order fractional-step time-discretization algorithm for solving the unsteady penetrative-convection model is studied. This equation is a nonlinear multi-physical model coupled by the unsteady incompressible Navier-Stokes equation and the heat conduction equation. The advantage of the algorithm is that the nonlinearity and incompressibility of the Navier-Stokes equations are separated to realize the high efficiency of the algorithm. Theoretically, under the assumption of the regularity of the solution, we obtain the first-order temporal convergence order of the velocity field and the temperature field. Finally, the convergence results are verified by numerical examples.
| [1] | Straughan, B. (2022) Effect of Anisotropy and Boundary Conditions on Darcy and Brinkman Porous Penetrative Convection. Environmental Fluid Mechanics, 22, 1233-1252. https://doi.org/10.1007/s10652-022-09888-9 |
| [2] | Carr, M. and de Putter, S. (2003) Penetrative Convection in a Horizontally Isotropic Porous Layer. Continuum Mechanics and Thermodynamics, 15, 33-43. https://doi.org/10.1007/s00161-002-0102-4 |
| [3] | Musman, S. (1968) Penetrative Convection. Journal of Fluid Mechanics, 31, 343-360. https://doi.org/10.1017/s0022112068000194 |
| [4] | Ravindran, S.S. (2012) Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model. Numerical Functional Analysis and Optimization, 33, 48-79. https://doi.org/10.1080/01630563.2011.618899 |
| [5] | Cao, M. and Li, Y. (2023) Optimal Error Analysis of Linearized Crank-Nicolson Finite Element Scheme for the Time-Dependent Penetrative Convection Problem. Communications on Applied Mathematics and Computation. https://doi.org/10.1007/s42967-023-00269-7 |
| [6] | Wan, W. and An, R. (2024) Convergence Analysis of Euler and BDF2 Grad-Div Stabilization Methods for the Time-Dependent Penetrative Convection Model. AIMS Mathematics, 9, 453-480. https://doi.org/10.3934/math.2024025 |
| [7] | Shen, J. (1992) On Error Estimates of Projection Methods for Navier-Stokes Equations: First-Order Schemes. SIAM Journal on Numerical Analysis, 29, 57-77. https://doi.org/10.1137/0729004 |
| [8] | Shen, J. (1996) On Error Estimates of the Projection Methods for the Navier-Stokes Equations: Second-Order Schemes. Mathematics of Computation, 65, 1039-1065. https://doi.org/10.1090/s0025-5718-96-00750-8 |
| [9] | Blasco, J. and Codina, R. (2004) Error Estimates for an Operator-Splitting Method for Incompressible Flows. Applied Numerical Mathematics, 51, 1-17. https://doi.org/10.1016/j.apnum.2004.02.004 |
| [10] | Wu, J.K., An, R. and Li, Y. (2021) Optimal H1 Error Analysis of a Fractional Step Finite Element Scheme for a Hybrid MHD System. Journal of Applied Analysis and Computation, 11, 1535-1556. https://doi.org/10.11948/20200277 |
| [11] | An, R. (2020) Error Analysis of a New Fractional-Step Method for the Incompressible Navier-Stokes Equations with Variable Density. Journal of Scientific Computing, 84, Article No. 3. https://doi.org/10.1007/s10915-020-01253-6 |
| [12] | Adams, R. (1975) Sobolev Spaces. Academic Press, New York. |
| [13] | Heywood, J.G. and Rannacher, R. (1990) Finite-Element Approximation of the Nonstationary Navier-Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization. SIAM Journal on Numerical Analysis, 27, 353-384. https://doi.org/10.1137/0727022 |
