求解对流扩散问题的积分方程法

doi:10.11949/j.issn.0438-1157.20150364

化工学报 ›› 2015, Vol. 66 ›› Issue (10): 3888-3894.DOI: 10.11949/j.issn.0438-1157.20150364

求解对流扩散问题的积分方程法

魏涛, 许明田, 汪引

山东大学土建与水利学院工程力学系, 山东济南 250061

收稿日期:2015-03-20 修回日期:2015-06-03 出版日期:2015-10-05 发布日期:2015-10-05
通讯作者: 许明田
基金资助:
国家自然科学基金项目(11272187)。

Integral equation approach to convection-diffusion problems

WEI Tao, XU Mingtian, WANG Yin

Department of Engineering Mechanics, Shandong University, Jinan 250061, Shandong, China

Received:2015-03-20 Revised:2015-06-03 Online:2015-10-05 Published:2015-10-05
Supported by:
supported by the National Natural Science Foundation of China (11272187).

摘要/Abstract

摘要：

提出了一种求解对流扩散问题的积分方程法。在这一方法中,首先利用Laplace方程的级数形式的格林函数将对流扩散方程转化为积分方程,然后利用级数的正交性质,把积分方程进一步简化为代数方程组,求解该方程组即可得到对流扩散方程的级数形式的近似解。最后,分别利用Chebyshev多项式和Fourier级数求解了3个典型的一维和二维对流扩散问题。该方法和有限体积法、有限元法和迎风差分法相比,展现出非常高的精度并且避免了由解的不连续性造成的虚假振荡。

关键词: 计算流体力学, 积分方程法, 对流扩散方程, 有限体积法, 传热

Abstract:

In the present work, an integral equation approach is developed to solve the convection-diffusion equations. In this approach, Green's function of the Laplace equation in the form of series is employed to transform the convection-diffusion equation into an integral equation. With the help of orthogonal polynomials, the integral equation is reduced to an algebraic equation system with a finite number of unknown variables. Finally, this integral equation approach is examined by three examples. The Chebyshev polynomial is used to approximate the one-dimensional convection-diffusion problem with nonhomogeneous boundary conditions and the Fourier series is for the two-dimensional convection-diffusion problem with homogeneous boundary conditions. The comparisons with the finite volume method, finite element method and upwind difference method show that the integral equation approach is more accurate and stable. The stability is also proved by the convection-dominated diffusion problems. Furthermore, it can achieve a satisfactory accuracy even with a small number of grid points.

Key words: computational fluid dynamics, integral equation approach, convection-diffusion equation, finite volume method, heat transfer

中图分类号:

O351.2

魏涛, 许明田, 汪引. 求解对流扩散问题的积分方程法[J]. 化工学报, 2015, 66(10): 3888-3894.

WEI Tao, XU Mingtian, WANG Yin. Integral equation approach to convection-diffusion problems[J]. CIESC Journal, 2015, 66(10): 3888-3894.

参考文献 28

[1]	Fudym O, Pradère C, Batsale J C. An analytical two-temperature model for convection-diffusion in multilayered systems: application to the thermal characterization of microchannel reactors [J]. Chem. Eng. Sci., 2007, 62(15): 4054-4064.
[2]	Li Huanyong, Gu Zhi, Zhang Haiyang, Li Wenwei. Thermodynamic analysis and growth of ZnSe single crystals in Zn-Se-I2 system [J]. J. Cryst. Growth, 2015, 415: 158-165.
[3]	Lü Hexiang (吕和祥), Qiu Kunyu (邱崑玉), Chen Jianfeng (陈建峰). Solution under convective coordinate for convection-diffusion chemical reaction kinetic equations [J]. CIESC Journal (化工学报), 2005, 56(1): 41-46.
[4]	Yang Xiaohua, Yang Zhifeng, Yin Xinan, Li Jianqiang. Chaos gray-coded genetic algorithm and its application for pollution source identifications in convection-diffusion equation [J]. Commun. Nonlinear Sci. Numer. Simulat., 2008, 13(8): 1676-1688.
[5]	Song Wei (宋伟), Kong Qingyuan (孔庆媛), Li Hongmei (李洪枚). Characteristics of VOC mass transfer from wood furniture surface [J]. CIESC Journal (化工学报), 2013, 64(5): 1549-1560.
[6]	Stynes M. Finite volume methods for convection-diffusion problems [J]. J. Comput. Appl. Math., 1995, 63(1/2/3): 83-90.
[7]	Sonar T. On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection [J]. Comput. Methods Appl. Mech. Engrg., 1997, 140(1/2): 157-181.
[8]	Farhloul M, Serghini Mounim A. A mixed-hybrid finite element method for convection-diffusion problems [J]. Appl. Math. Comput., 2005, 171(2): 1037-1047.
[9]	Stynes M. Steady-state convection-diffusion problems [J]. Acta Numerica, 2005, 14: 445-508.
[10]	Tian Z F, Ge Y B. A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems [J]. J. Comput. Appl. Math., 2007, 198(1): 268-286.
[11]	Shih Y, Cheng J Y, Chen K T. An exponential-fitting finite element method for convection-diffusion problems [J]. Appl. Math. Comput., 2011, 217(12): 5798-5809.
[12]	Ge Yongbin, Cao Fujun. Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems [J]. J. Comput. Phys., 2011, 230(10): 4051-4070.
[13]	Zingg D W, De Rango S, Nemec M, Pulliam T H. Comparison of several spatial discretizations for the Navier-Stokes equations [J]. J. Comput. Phys., 2000, 160(2): 683-704.
[14]	de Rango S, Zingg D W. Higher-order spatial discretization for turbulent aerodynamic computations [J]. AIAA Journal, 2001, 39(7): 1296-1304.
[15]	van Leer B. Towards the ultimate conservative difference scheme (Ⅴ): A second-order sequel to Godunov's method [J]. J. Comput. Phys., 1979, 32(1): 101-136.
[16]	Venkatakrishnan V. On the accuracy of limiters and convergence to steady state solutions [R]. Reno, NV: AIAA, 1993.
[17]	Venkatakrishnan V. Convergence to steady state solutions of the Euler equations on unstructured grids with limiters [J]. J. Comput. Phys., 1995, 118(1): 120-130.
[18]	Jiang Guangshan, Shu Chiwang. Efficient implementation of weighted ENO schemes [J]. J. Comput. Phys., 1996, 126(1): 202-228.
[19]	Friedrich O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids [J]. J. Comput. Phys., 1998, 144(1): 194-212.
[20]	Hu Changqing, Shu Chiwang. Weighted essentially non-oscillatory schemes on triangular meshes [J]. J. Comput. Phys., 1999, 150(1): 97-127.
[21]	Wen P H, Hon Y C, Li M, Korakianitis T. Finite integration method for partial differential equations [J]. Appl. Math. Model., 2013, 37(24): 10092-10106.
[22]	Hernandez-Martinez E, Puebla H, Valdes-Parada F, Alvarez-Ramirez J. Nonstandard finite difference schemes based on Green's function formulations for reaction-diffusion-convection systems [J]. Chem. Eng. Sci., 2013, 94: 245-255.
[23]	Xu Mingtian, Stefani F, Gerbeth G. Integral equation approach to time-dependent kinematic dynamos in finite domains [J]. Phys. Rev. E, 2004, 70(5): 056305.
[24]	Xu Mingtian, Stefani F, Gerbeth G. The integral equation approach to kinematic dynamo theory and its application to dynamo experiments in cylindrical geometry [J]. J. Comput. Phys., 2008, 227(17): 8130-8144.
[25]	Xu Mingtian, Wei Tao, Zhang Longzhou, Hao Aiqin. An integral equation approach to two-dimensional incompressible resistive magnetohydrodynamics [J]. Geophys. Astrophys. Fluid Dyn., 2014, 108(4): 463-478.
[26]	Xu Mingtian. Effect of soft-iron impellers on the von Kármán-sodium dynamo [J]. Phys. Rev. E, 2014, 89(1): 013012.
[27]	Versteeg H K, Malalasekera W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method[M]. 2nd ed. London: Pearson Prentice Hall, 2007: 115-155.
[28]	van Niekerk F D, van Niekerk A. An optimal rational basis function in the finite element method for convection-diffusion problems in one space variables [J]. Mathl. Comput. Modelling, 1994, 19(5): 43-50. 013012.
[27]	Versteeg H K, Malalasekera W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method[M]. 2nd ed. London: Pearson Prentice Hall, 2007: 115-155.
[28]	van Niekerk F D, van Niekerk A. An optimal rational basis function in the finite element method for convection-diffusion problems in one space variables[J]. Mathl. Comput. Modelling, 1994, 19(5): 43-50.

求解对流扩散问题的积分方程法

Integral equation approach to convection-diffusion problems

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