The Solution and Numerical Simulation of Several Three Dimensional Convection Diffusion Equations

摘要

In this paper, the instantaneous point source diffusion model, the continuous time source diffusion time-varying model and the continuous time arbitrary shape source diffusion time-varying model in convection-diffusion equation are studied. First, by establishing the appropriate model assumptions, under certain initial conditions, the analytical solution of instantaneous point source diffusion model is obtained by Fourier transform. Using this result can get the solution of the source diffusion model at the continuous time point by integrating the result in time. The volume of the source shape and the integral of the time are obtained, and the solution of the diffusion model with arbitrary shape in continuous time is obtained, a series of graphs (chamfer, gradient, contour, etc.) describing the diffusion characteristics were drawn; and the solution images of each model were analyzed and compared. This article studies the instantaneous point source diffusion time-varying model, the continuous time point source diffusion time-varying model and the continuous time shape source diffusion time-varying model. A proper model assumption is established firstly. Under the given initial conditions, the analytical solution of instantaneous point source diffusion model is obtained with Fourier transform. The solution of the continuous time point source diffusion time-varying model is obtained with the integral in time domain. And the solution of the continuous time shape source diffusion time-varying model is obtained with the triple integral in the shape of the source and then the integral in time domain. Then, the numerical simulation of these models above is given with the help of MATLAB, drawing a series of images which describe the characteristics of the diffusion (The sectional drawing, gradient map, contour map and so on). The analysis and comparison of these images are given.