A Modified Finite Integration Method For Convection Diffusion Problems

The convection-diffusion problems often appear in various scientific and engineering problems which involve in many disciplines,such as,biology,physics and chemistry.Numerical methods are widely employed to solve the convection-diffusion problems because of the difficulty to obtain the exact solutions.However,the numerical solutions of many numerical methods demonstratenonphysical spurious oscillation for the convection-dominatedproblems.Therefore the key point for convection-diffusion problemsis to develop numerical methodswhich maintain high accuracy,high stability and high convergence when dealing with the convection-dominated convection diffusion problems.Recently,the finite integration method has been proposed to solve the partial differential equations.In this thesis,a new modified finite integration method is developed to solve the convection-diffusion problems.For using the finite integration method to solve the one dimensional convection diffusion problem,the governing equation is firstly integrated,andall the derivatives of the unknown function is eliminated,thenthe Simpson'srule is appliedto approximate the resulting integrals.Numerical results show that this method can achieve a high accuracy evenif several nodes are employed to divide the computational domain for the diffusion dominated convection diffusion problems.In order to improve the stability and accuracy of the finite integration method in dealing with the convection-dominated transport problems,a weight coefficient is introduced when discretizing the integrals.This weight coefficient can be adjusted to account for the flow direction.The modified finite integration method shows better accuracy and stability than the traditional finite volume method for the convection-dominant transport problems.For the unsteady convection-diffusion problems,the C-N method and fully implicit method which are widely applied in the traditional finite volume method are employed to discretize the time variable.The fully implicit method shows a better stability and convergence than the C-N method in solving the unsteady convection-diffusion problems with source.In extending the modified finite integration method to the two dimensional convection-diffusion problems,the discrete matrix becomes more complicated and more unknown functions appear.These unknown functions are constructed by interpolation method.The unsteady two dimensional convection-diffusion problemsare solved by thefully implicit method for the time variable.The modified finite integration method shows better accuracy and stability than the traditional finite volume method.