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Research On The Algorithm Of Convection-dominated Diffusion Of Two Dimensions Based On Characteristic Theory

Posted on:2010-10-07Degree:MasterType:Thesis
Country:ChinaCandidate:N B HuFull Text:PDF
GTID:2120360275484407Subject:Basic mathematics
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The Convection-diffusion equations are the basic equations, and can be used to describe many physical phenomena such as the distribution of pollutants in the river, air and nuclear waste pollution, the flow of fluid and the transfer of heat in fluid and so on. It has a common character, namely, dominated convection. However, the character has brought much difficulty for numerical solution. Therefore, finding an efficient numerical solution is an important topic in the field of computational mathematics. During 1980's, Douglas and Russell, etc., put forward the characteristic correct method to solve convection-dominated diffusion problems, which could combine with other methods, and give some methods such as characteristic finite differ- ence method and characteristic-mixed finite element method, etc., and give them theoretical analysis.In terms of the fact that most works focus on the situation of one dimension, we choose the linear convection-dominated diffusion of two dimensions as our model in this paper. The major contributions of this thesis are in two aspects:Firstly, we will give the discrete format of equation by the expanded characteristic-mixed finite element method to numerically solve the model problem. This scheme can simultaneously deal with three variables: the scalar unknown, its gradient and its flux optimally to approximate precisely. That is, we adopt expanded mixed finite element method to treat the term of diffusion. While we discretize the convection part along with direction of the characteristic line in order to overcome the phenomena of numerical oscillations at the fluxion. Thus, it can ensure the stability of this algorithm. By the theoretical analysis, for the two dimensional case, this scheme is stable and attains an optimal L2 error estimate. Secondly, we simulate it by adopting characteristics finite difference method under the same model but with different boundary conditions. The authors of [27] discuss the example of one dimension. We find the bilinear interpolation to solve the convection-dominated diffusion equation of two dimensions by further analysis, give its discrete scheme, and theoretically analyze the convergence of this scheme, which shows the error of order (O h 2+Δt ) . And then, we show a numerical example by MATLAB 7.0. By the comparison of the figure, it is shown that the difference scheme can elimi- nate the numerical oscillations more efficiently for a kind of convection- dominated diffusion of two dimensions. Thus, we can improve the approxi- mation rate of numerical solution.
Keywords/Search Tags:Linear convection-diffusion equation of two dimensions, Expanded characteristic-mixed finite element method, Characteristic finite difference method, Bilinear Interpol- ation, Error estimates, Convergence analysis
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