The Study And Application Of MWLS Algorithm For Solving Parabolic Partial Differential Equations

Parabolic partial differential equations are important math-physics equations which describe the process such as the heat conduction, the diffusion, the seepage and so on. Traditional numerical methods are chief ways. But, their disadvantages of the mesh restrained expose increasingly. So searching for new algorithms is very necessary. Meshless method is a focus in numerical analysis at home and abroad these years, and then it may conquer mesh dependence to eliminate mesh completely or partially. Meshless weighted least-square method (MWLS) is one of new meshless methods. In the thesis, MWLS algorithm and application on parabolic equations are studied.The basic theory of meshless methods is introduced by the numbers. This part gives a presentation of the moving least-square approximation in detail, studies the factors of its error influencing by the numerical examples, and discusses the convergent problems of the least-square method. MWLS being applied straight to parabolic equations is difficult to solve problems because of complex time integral. MWLS is combined with finite difference method to solve heat conduction and convection diffusion equations, and differential-MWLS algorithm is structured in the paper. The numerical examples show that the algorithm may have lesser computational complexity and attain higher accuracy. Comparing with FEM, the algorithm gets rid of mesh astriction in space region, has the advantages of higher accuracy and is convenient in dealing with in the front or behind. Its program is easy. So the method is a new and efficient numerical method.In this thesis, relating numerical examples discuss the main factors, which may affect the calculation precision in the two aspects: constructing approximate functions and dispersing differential equations. Then some suggestions, which can get the best results, are given. The bigger influencing error is the radius of influence in MLS. The errors have closely relation with the length of the time step and the ways of dispersing equations. And MWLS is better in accuracy and efficiency than direct collocation method and Galerkin method.