Research On A Boundary Type Method For Solving Convection-Diffusion Equations

Convection-diffusion equation is a kind of basic motion equation,which includes the diffusion item and the convection item.It can be used to describe the distribution of pollutants in river pollution,air pollution,nuclear pollution,fluid flow,heat convection in fluid and many other physical phenomena.But for this kind of equation,except for a few simple cases,the exact solution of most of the problems can not be obtained at present.Therefore,using the numerical method is the main method to carry out the numerical simulation.The construction of accurate,stable and efficient numerical method has become an important content of the research.In this thesis,a boundary type method,half boundary method,is proposed to numerically solve the linear and nonlinear convection-diffusion equations.Through the comparison with other numerical methods,the advantages of half boundary method in results precision and solving efficiency are presented.The main idea of half boundary method is to reduce the order of the differential equation by introducing mixed variable.Then the relation of variables at the adjacent nodes can be obtained through the integral operation and then the relation of the variables at any node in the domain and the variables at half of the boundaries can be obtained through derivation.After the unknown variables at half of the boundaries are solved by the boundary conditions,the variables at any node can be obtained by the relation.The unknown variables in this method only exist at half of the boundaries,which means it is a boundary type method.It should be noticed that although the half boundary method belongs to the boundary type method,it is completely different from the traditional boundary element method.When establishing the discrete equation,it does not need a basic solution,but directly establishes the differential equation with mixed variables.Compared with the finite volume method,the half boundary method has fewer unknown variables when the grid number is identical.And the calculation matrix involved in the solution of one-dimensional problem is only of second order,so it is unnecessary to solve the large matrix equation.Therefore,the calculation amount is reduced and the required computational memory is reduced.In addition,the newly introduced unknown variable has physical significance,which enables to obtain the global solution by the half boundary method.For the convection-diffusion equation,the convection-dominated problem has always been one of the problems worthy of attention.Such problems have hyperbolic properties,and their solution functions have boundary layers with large gradient changes.Moreover,the larger the Peclets number is,which expresses the convection domination,the narrower the boundary layer region is.When solving such problems,the traditional numerical calculation methods may have numerical oscillations in the boundary layer region and can not obtain accurate numerical result.For the convection-dominated problem,under the condition of the same node number,a more accurate numerical solution can be obtained by using the half boundary method compared with the finite volume method.In addition,by reducing the local Peclet number using the nonuniform grid,more accurate numerical trsults can be obtained on the basis of ensuring the computational efficiency.For convection-dominated problems,the half boundary method has advantage in calculating accuracy.Besides,many researches are based on the constant coefficient model,which is very convenient for solving.However,in practical engineering problems,the coefficients in the convection-diffusion equation are not constant usually,and even discontinuous coefficients may appear.Discontinuous coefficient is a special case of variable coefficient problem.Due to the discontinuity,the condition for continuity must be considered in many other numerical methods.As a result,the number of equations to be solved increases,the overall matrix increases and the computational efficiency decreases.But for the half boundary method,there is no need of setting continuity conditions at the discontinuous position,and the numerical results show no numerial oscillation.This characteristic makes half boundary method has advantage for solving problems with discontinuous model.The convection-diffusion equations applied in practical problems are not all linear.The coefficients of the equations may have a certain relationship with the solutions,so the form of nonlinear convection-diffusion equations will appear.The Burgers equation is a typical nonlinear convection-diffusion equation with nonlinear convection item,which can be used as the model equation of the incompressible Navier-Stokes equation.Because of the complexity of nonlinear equation,the nonlinearity of the convection item often makes the solution produce a sharp gradient in some regions,so it is more difficult to obtain the numerical solution.For nonlinear convection-diffusion problems,the iteration algorithm is introduced in half boundary method,through which the nonlinear coefficient is approximated using the iteration value.Other steps are the same with the linear equation solving procedure.The numerical examples of nonlinear convection-diffusion equation prove that using half boundary method can obtain accurate convergence solution.Considering the convection-dominated cases of nonlinear convection-diffusion equations,the non-uniform grid is applied to obtain stable and highly precise numerical solution.