An Improved Spectral Method For Convection Diffusion Equations And Vorticity Stream-function Equations

Convection diffusion equation and vorticity stream-function equation are two types of equations commonly used in fluid mechanics.Convection diffusion equation is also widely used in heat transfer,chemistry and environmental science.Vorticity stream-function equation plays an important role in studying the flow of viscous incompressible fluids.In practical problems,due to the complexity of the solution region and boundary conditions,it is difficult to obtain analytical solutions for these two types of equations.Solving them numerically has become an important method.Therefore,constructing efficient and stable numerical methods has important application value.Based on the idea of spectral method and integral equation theory,an improved spectral method is proposed to numerically solve the convection diffusion equation and vorticity stream-function equation.When using the improved spectral method to solve the steady convection diffusion problem with the first kind of homogeneous boundary conditions,the first derivative in the convection term can be eliminated by variable substitution.Then the auxiliary boundary value problem about Green’s function is introduced.The Green’s function and the unknown function are expanded by sine series,and the expression of Green’s function is obtained.Then,according to the properties of Green’s function,the convection diffusion equation is transformed into an integral equation.Finally,by using the orthogonality of sine series,the integral equation containing spatial variables and series coefficients can be simplified into a system of integral equations containing only series coefficients.By solving this system of integral equations,the approximate solution of convection diffusion equation can be obtained in terms of finite sum of sine series.Several examples of one-dimensional and two-dimensional steady convection diffusion problems are solved by using the improved spectral method.The calculation results show that this method has higher accuracy and does not require meshing in comparison with the finite volume method and the QUICK scheme.Compared with the integral equation method,the derivation and programming process are simplified on the premise of ensuring the accuracy.For the unsteady convection diffusion problem,it involves not only spatial variables,but also time variables.The spatial variables can be discretized in the same way as in the previous steady state case,and the integral equation about time and space variables can be simplified into a system of integral equations only about time variable.The fourth-order Runge-Kutta method is employed to solve the system of integral equations,and the approximate solution of the original problem in the form of finite sum of series can be obtained.Finally,several examples of one-dimensional and two-dimensional unsteady convection diffusion problems are used to examine the improved spectral method,and the effects of Peclet number,the number of truncation terms,time step,computation time and different source terms on the computational error are discussed.For the nonlinear vorticity stream-function equation,the auxiliary initial boundary value problem about Green’s function is introduced,and the vorticity stream-function equation is transformed into an integral equation by using the properties of Green’s function.Then the unknown functions and Green’s functions under the periodic boundary conditions and the first kind of homogeneous boundary conditions are expanded into Fourier series and sine series,respectively.Then the integral equation is transformed into a system of integral equations with only time variable by using the orthogonality of series,and the fourth-order Runge-Kutta method is used to solve this system of integral equations,and the approximate solution of the original equation in terms of the finite sum of the series is obtained.Finally,the improved spectral method is tested with several examples under different boundary conditions,and it is found that the computational results are in good agreement with the analytical solutions.