Compact Finite Volume Method For 1D Unsteady Convection Diffusion Equations

In this paper,we consider the one dimensional convection-diffusion equation with constant coefficients and first boundary conditions:where f(x,t)??(x)?g1(t)?g2(t)?a(x)?p(x)?q(x)satisfies the periodic boundary conditions,a(x)has a positive lower bound,q(x)>0.The convection-diffusion equation is an important partial differential equa-tion.It can describe distribution of the pollutants in river pollution,flowing of the fluid in reservoir simulation,the transmission of electrons in a semiconduc-tor device and others physical phenomenon.The effective numerical solution of convective diffusion problem has been the important content in numerical math-ematics.The mainly numerical method of solving convective diffusion equation has finite difference methods,finite element methods,finite volume methods and others methods.Especially when convection is dominant,the traditional method will appear numerical oscillation and numerical dispersion.For the case when convection is not dominant,The integral form of conser-vation law is derived by integrating the equation over control volumes.Then the compact finite volume scheme is obtained by discretizing the integral form based on Taylor formula and quadratic Lagrange interpolation.Consequently the compact difference volume scheme and the norm error estimation of the numerical solution is given.The numerical experiments show this scheme has satisfying performance.For the case when convection is dominant.We combines the compact upwind method with the compact finite volume method,then integrates the function on the controlled volume and discretizing the integral form based on Taylor formula and quadratic Lagrange interpolation.In order to eliminate the numerical fluctua-tion caused by convection dominance,the convective part adopts compact upwind method.Not only does this scheme have the character of high precision,but also the matrix of the deduced liner algebraic system is tridiagonal,which can be easily solved.Finally,the truncation error and the stability of the scheme are given.The numerical experiments show this scheme has satisfying performance.