The Fast Boundary Element Algorithm For Two Dimensional Unsteady Convection Diffusion Problem

The convection-diffusion model is a very basic mathematical and physical model that can be applied to many fields such as river slope protection,air quality monitoring,underground oil and natural gas extraction,and so on.Therefore,it is of great practical significance to study the numerical solution of the convection-diffusion model.At present,classical methods such as finite difference method(FDM),finite element method(FEM)and finite volume method(FVM)can be used to solve the convection-diffusion problem.However,for the problem of convection dominance,these classical methods will produce severe numerical dissipation and oscillation during calculation.Although it is possible to avoid numerical shocks and obtain stable solutions by encrypting the grid,it will lead to a large-scale discrete linear system,which will cause difficulties in calculation,storage,and solution.If the boundary element numerical method is used to analyze such convection diffusion problems,it is very valuable to obtain a low-order calculation model that can approximate the characteristics of the original model with a small amount of work.In this paper,based on the boundary element method,the model orthogonal analysis of the two-dimensional unsteady convection diffusion problem is carried out using the idea of orthogonal decomposition of features.The main work includes:(1)A two-dimensional unsteady convection diffusion problem based on radial integration method is studied.First,by using the basic solution of the Laplace equation,the boundary-domain integral equation of the two-dimensional unsteady convection diffusion problem is established by the weighted residual method.Secondly,for the domain integration with unknown quantity,the unknown quantity is approximated by the fourth-order spline basis function,and then the domain integration is converted into boundary integral by radial integration method.Finally,a pure boundary element algorithm that only requires boundary discretization and some internal point representations is formed.(2)Based on the boundary element method,a reduced-order model of the two-dimensional unsteady convection diffusion problem is established.First,reorganize the BEM discrete format obtained in(1)to form a system of first-order ordinary differential equations with uniform variables.Secondly,the numerical solution of the boundary element at some moments is taken as the instant image matrix,and the orthogonal orthogonal matrix is subjected to characteristic orthogonal decomposition to establish the POD mode.Finally,the POD modal matrix is used to reduce the order of the original first-order ordinary differential equations,and a reduced-order model is established and solved.(3)Three numerical examples with different boundary conditions and initial conditions are selected to study the influence of factors such as different peclet numbers,different time steps,and different boundary element grid models on the applicability and effectiveness of the method described in this paper.