PERTURBATIONAL FINITE VOLUME METHOD FOR CONVECTIVE DIFFUSION EQUATION AND DISCUSSION

A perturbational finite volume (PFV) method for the convectivediffusion equation is presented in this paper. PFV method uses first-orderupwind scheme as its starting point, the mass fluxes of the cell faces aremodified by a numerical-value perturbation technique i.e. the mass fluxesare expanded into power series of the grid spacing and the coefficients ofthe power series are determined withthe aid of the conservation equation itself. The resulting formulae of theabove perturbation operation are higher-order upwind and central PFVschemes. They include the second-, third-, and fourth-order upwind PFVschemes 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 theconvective boundedness criteria, they do not produce oscillatorysolutions, expecially their numerical diffusions are much smaller thanthose of the first-order upwind scheme. The central PFV schemes withsecond- and fourth-order accuracy are positive grid-centered FV ones forany values of the grid Peclet numbers and then are more better than thenormal second-order central FV scheme.Two numerical examples (including a lid-driven cavity flow and problem ofscalar quantity transport in the one-dimensional flow) are computed toillustrate 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.