Integral equation approach to convection-diffusion problems

doi:10.11949/j.issn.0438-1157.20150364

Abstract

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

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

CLC Number:

O351.2

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

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.

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