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Some Compact Finite Difference Methods For Convection-diffusion Problems

Posted on:2010-05-31Degree:MasterType:Thesis
Country:ChinaCandidate:X Y YangFull Text:PDF
GTID:2120360272499674Subject:Computational Mathematics
Abstract/Summary:PDF Full Text Request
Convection-diffusion equation is one of the basic equations in the field of fluid mechanics. The solving of this equation is of great significance in theoretical research and in practical application. For convection-dominated flow, many numerical methods often give pseudo solution or non-physical negative solution. At present, common numerical methods for stationary convection-dominated equations are: finite element method, finite volume method, finite analytic method and finite difference method. Among these, the finite difference method is widely studied and used for the sinplicity of the scheme and forming the linear algebraic equations easily. In the finite difference method, the compact high-accuracy difference scheme especially becomes a research hotspot for it involves a fewer grid points and the boundary should not be treated specially, and the numerical result has a better accuracy.In this paper, we consider stationary linear convection-diffusion equation with source. We try to design new compact difference schemes. The coefficients of the scheme will be determined by Taylor expansion,that is,every point around the center point will be expanded by using Taylor format, but different from the traditional finite difference method, we will keep infinite number of terms instead of finite terms (these terms constitute a convergent scries),in order to obtain the better accuracy of the scheme . Meanwhile,during the construction of the scheme, we will point out the constrain of traditional difference scheme and the reason why it is so difficult to construct the scheme with high accuracy.In the second section of this paper, one-dimensional linear convection-diffusion equation with the source is considered. First,an unconditionally stable three-point compact difference is constructed which has second-order accuracy, Numerical experiments show that the calculation accuracy of the scheme is better than many other second-order schemes and even fourth-order schemes given by predecessors. Then a third-order conditionally stable scheme is given through coefficient amendment. A fourth-order scheme through coefficient perturbation is given at last. Numerical experiments for all the schemes are provided.In the third section,we will obtain a second-order five-point difference scheme for two-dimensional problems by applying the second-order basic scheme in section one. Then a fourth-order nine-points scheme is given through coefficient perturbation. Finally a five-point new scheme with unconditionally stable is constructed by Taylor expansion and convergence of series. Numerical examples will reveal the efficiency of all these schemes.
Keywords/Search Tags:convection-diffusion equation, compact difference scheme, convection dominating problem, singular perturbation
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