Application Of Finite Integral Method For Convection Diffusion Problems

Convection and diffusion are the main factors that dominate the flow and transport.The continuity equation,momentum conservation equation and energy equation in hydrodynamics can be regarded as cornvection-diffusion equation.Therefore,it is of great value to study the numerical simulation method of convection-diffusion equation.Firstly,Finite Volume Method(FIM)is improved and then applied to solve the convection-diffusion problem.In the application of FIM,the partial derivatives of the governing equation are eliminated by integrating the governing equation in the region of fluid flow.Then the boundary conditions are brought into the governing equation.Finally,the integral equation is discretized into algebraic equations by using the numerical integral method.In this paper,FIM is improved to reflect the transport of flow,simplify the treatment of boundary conditions,and can be easily extended to solve two-dimensional even multi-dimensional convection-diffusion problems.The numerical experiments show that compared with the traditional Finite Volume Method(FVM),FIM has obvious advantages in solving one-dimensional and two-dimensional steady convection-diffusion problems and unsteady convection-diffusion problems:it keeps good stability in solving convection-dominated flow problems.In some examples,the numerical solution of FIM based on trapezoidal integral formula is even better than that of quadratic upwind QUICK scheme.This paper presents a way to construct discrete integrals with different integral independent variables to form algebraic equations,which is used to deal with the integration of non-constant terms in the governing equation.Taking the steady diffusion problems in polar coordinates as examples,the basic method of solving non-constant partial differential equations by FIM is pointed out.It can be further studied and developed to solve more complex partial differential equations and flow problems.In this paper,the effects of integral method,mesh generation,boundary conditions and higher order interpolation integration on the numerical solution of convection-diffusion equation by FIM is analyzed.It is proved that Finite Integral Method(FIM)with equation analysis and reasonable optimization design has higher computational efficiency than Finite Volume Method(FVM),while the convergence condition is more relaxed,and the convergence speed and the result precision are satisfactory in solving unsteady problems at the same time.Several integration methods and node distributions are attempted to solve steady diffusion problems in polar coordinates,which point that grid settings need to be adjusted to ensure the accuracy and stability of the calculation for the discontinuous integral independent variables.Finally,the approach of Finite Integral Method(FIM)proposed in this paper can be further improved through adaptive grids,different integral functions and attempts to solve more complex flow problems.