A Finite Integration Method For Convection Diffusion Problems

Convection-diffusion equation finds wide applications in various scientific and engineering problems.It can be employed to describe many phenomena,such as,the diffusion of polluted substances in water,air,the diffusion of heat and salinity in the ocean and so on.Because of the complex structures of the heat transfer media and the diversity of the boundary conditions,the analytic solutions of the convection-diffusion equations are not available.The numerical simulation is a powerful tool for solving the convection-diffusion problems.Therefore,it is of great importance to develop efficient numerical methods to solve the convection-diffusion problems.Recently,a finite integration method was developed for solving partial differential equations.In this method,the partial differential equation is firstly integrated into a pure integration equation which is solved by numerical quadrature.In the present work,the finite integration method is extended to solve the convection-diffusion equations.For the diffusion-dominated convection-diffusion problems,numerical results show that the finite integration method is more than one and two orders of magnitude better in accuracy than the QUICK scheme and the central differencing scheme,respectively.Furthermore,a finite integral method for two dimensional convection-diffusion problems is developed,and the effectiveness of the method in dealing with the two-dimensional convective-diffusion problem is demonstrated by an example.A finite integral method is also established for solving the unsteady convective-diffusion problem.It is well known that numerical simulation of convection-dominated flow phenomena remains one of the most challenging problems in computational fluid dynamics.We find that the finite integration method can be improved by introducing a weighting parameter in the numerical quadrature.By appropriately adjusting this parameter,the modified finite integration method is more than four orders of magnitude better in accuracy than the traditional finite volume method for the convection-dominated flows.Furthermore,even coarse grids are utilized to divide the flow domain the modified finite integration method has not lead to unphysical oscillations for the convection-dominated convection-diffusion problem,which demonstrates a good stability of the modified finite integration method.The improved finite integral method has a wide application prospect in numerical simulation of convective diffusion problem.