Study On Two Numerical Methods For Solving A Class Of Convection Diffusion Equation(s)

Convection-diffusion equation(s)have been used in the fields of mechanics,physics and environmental science.They can describe some diffusion phenomena such as mass,heat transfer process,pollutant distribution in water and so on.Because the convection-diffusion equation(s)is difficult to obtain the analytical solution,and the traditional numerical methods have shown their advantages in the process of solving the convection-diffusion equation(s),so the study of numerical methods still has a certain significance for solving the convection-diffusion equation(s).This paper mainly studies two kinds of numerical methods with high precision,namely Fourier spectrum method and barycentric Lagrange interpolation collocation method.Four(1+1)dimensional convection-diffusion equations,one(1+1)dimensional fractional convection-diffusion equations,two(2+1)dimensional convection-diffusion equations and two(2+1)dimensional fractional diffusion equations are numericallysimulated by different initial conditions and boundary conditions.Comparison with other numerical methods and numerical results show that these two methods have high precision,and the effectiveness of Fourier spectrum method and barycentric Lagrange interpolation collocation method is illustrated.In particular,the Fourier spectrum method is used to simulate the(2+1)dimensional fractional diffusion equations.Some patterns are obtained by giving different initial conditions and parameters.The reliability of the numerical results is proved,which shows that the Fourier spectrum method is not only effective for the integer order convective diffusion equations.The numerical results for the fractional diffusion equation are also good.Secondly,the meshless method is applied to the actual model,that is,the meshless method is used to solve the one-dimensional nitrogen replacement model.Through analysis and comparison,the factors affecting the nitrogen concentration distribution can be obtained,and it is known that pipelines with different diameters of materials will have an impact on the nitrogen concentration distribution.In addition,the nitrogen concentration distribution will change under the influence of different time and different turbulence diffusion coefficient,which has a certain significance in practical application and lays a foundation for future research.