对流扩散方程的摄动有限体积(PFV)方法及讨论

PERTURBATIONAL FINITE VOLUME METHOD FOR CONVECTIVE DIFFUSION EQUATION AND DISCUSSION

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.

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.