PERTURBATIONAL FINITE VOLUME METHOD FOR CONVECTIVE DIFFUSION EQUATION AND DISCUSSION
对流扩散方程的摄动有限体积(PFV)方法及讨论

A perturbational finite volume (PFV) method for the convective diffusion equation is presented in this paper. PFV method uses first-order upwind scheme as its starting point, the mass fluxes of the cell faces are modified by a numerical-value perturbation technique i.e. the mass fluxes are expanded into power series of the grid spacing and the coefficients of the power series are determined with the aid of the conservation equation itself. The resulting formulae of the above perturbation operation are higher-order upwind and central PFV schemes. They include the second-, third-, and fourth-order upwind PFV schemes as well as the second- and fourth-order central PFV schemes. The properties of PFV schemes are discussed and proved. The second-, third- and fourth-order upwind PFV schemes satisfy the convective boundedness criteria, they do not produce oscillatory solutions, expecially their numerical diffusions are much smaller than those of the first-order upwind scheme. The central PFV schemes with second- and fourth-order accuracy are positive grid-centered FV ones for any values of the grid Peclet numbers and then are more better than the normal second-order central FV scheme. Two numerical examples (including a lid-driven cavity flow and problem of scalar quantity transport in the one-dimensional flow) are computed to illustrate excellent behaviors of PFV schemes. In addition, a conceptional comparison of PFV scheme is also given with the perturbational finite difference scheme, multinodes and compact schemes.