A Study On Finite Point Method For Nonlinear Convection Diffusion Equation

The meshless method is a kind of numerical calculation method developed in recent years.The method adopts the point-based approximation and does not need to establish the grid, thus overcoming the dependence of the traditional method on the grid, applying well to high-speed collision and penetration, fluid mechanics and other issues to solve. Therefore, the advantages and application prospects of this method is prominent in many fields.The convective diffusion model is mainly used in many fields such as fluid mechanics,aerodynamics, environment and financial engineering. Numerical solution of convection diffusion equation has been an important subject in the field of research. Aiming at the convection diffusion equation with numerical oscillations, this paper uses Onate’s finite point method for meshless stability to apply the steady-state term before the discrete equation, so as to avoid the oscillation generated by solving the equation. The method is applied to the linear and nonlinear convection diffusion equations. The main research work is as follows:In this paper, the history of meshless development in recent years is firstly summarized,the finite point method and the current research status is introduced, and then the principle of stabilizing the finite point method is analyzed and deduced, including the application of the stationary term, the moving least squares construction approximation function , the selection of the weight function. the influence of the size of the support domain on the accuracy of the approximate function and the discrete scheme of the equation. The main factor that influence the error is the selection of domain factor scale by the analysis of moving the least squares curve approximation.Secondly, the finite point algorithm is used to derive and simulate the one-dimensional and two-dimensional linear convection diffusion equations generated by the actual physical background. The relationship between the calculation results and the support domain size, step size and time is discussed in depth. The result shows that the algorithm has the characteristics of simplicity, stability and efficiency. Compared with the traditional finite element method and finite difference method, this algorithm can attain an equal and even higher calculation accuracy.Besides, And compared with the precise solution of the figure, the method can eliminate the oscillations. Thus, it proves that the method is an effective numerical method to solve the nonlinear convection diffusion equation.Finally, based on the linear equation, the meshless finite point method is applied to the nonlinear convection diffusion equation, and nonlinear processing is carried out with simple forward iterations. Since the meshless finite point method is required to apply stable term processing, the diffusion term and the resource term with nonlinear terms must be considered in this paper. Then the finite point algorithm is deduced in this case. Thus, the corresponding numerical simulation is given to discuss the relationship among calculation result, support domain size, step size, and time. Compared with the traditional finite element method and finite difference method, this algorithm can attain a higher calculation accuracy. Therefore, it proves that the finite element method is also an effective numerical method to solve the nonlinear convection diffusion equation.