对流扩散方程的变步长摄动有限差分格式

摄动有限差分(PFD)方法是构造高精度差分格式的一种新方法。变步长摄动有限差分方法是等步长摄动有限差分方法的发展和推广。对需要局部加密网格的计算问题，变步长PFD格式不需要对自变量进行数学变换，且和等步长PFD格式一样，具有如下的共同特点：从变步长一阶迎风格式出发，通过把非微商项(对流系数和源项)作变步长摄动展开，展开幂级数系数通过消去摄动格式修正微分方程的截断误差项求出，由此获得高精度变步长PFD格式。该格式在一、二和三维情况下分别仅使用三、五和七个基点，且具有迎风性。文中利用变步长PFD格式对对流扩散反应模型方程，变系数方程及Burgers方程等进行了数值模拟，并与一阶迎风和二阶中心格式及其问题的精确解作了比较。数值试验表明，与一阶迎风和二阶中心格式相比，变步长PFD格式具有精度高，稳定性与收敛性好的特点。变步长PFD格式与等步长PFD格式相比，变步长PFD解在薄边界层型区域的分辨率得到了明显的提高。

The perturbational finite difference(PFD) scheme of variational step of convection-diffusion equation

The perturbational finite difference method (PFD) is a new method of constructing high accurate difference scheme. The PFD of variational step is a development and extension of PFD of uniform step. The PFD of variational step dose not require any mathematical transformation of independent variables when dealing with problems that require local refinement of computing grids, moreover, it shares the same characteristics with PFD of uniform step as follows: Consider the first order upwind scheme of vanatiorial step as the starting point, the non- differential terms (convection coefficient and source term) were expanded into power series of grid spacing and the coefficients of the power series were determined hy means of elimina ting the truncation error of the corresponding modified differential equation, and thus we obtain the high accurate PFD scheme of vanational step. This scheme uses three, five and seven nodes under the circumstances of one, two and three dimensional prohlems, and also the scheme is an upwind scheme. This article employed the PFD scheme of vanational step to calculate the convection-diffusion equation, the equation of variational coefficient and Burgers equation and so on, and a comparison with the normal fist order upwind and second order central scheme was made. The results of the numerical experiments show that the PFD scheme of vanational step has higher accuracy and better stability and convergence than that of the fist order upwind and second order central scheme. The resolution of the PFD scheme of vanational step has been clearly improved when it deals with prohlems which have thin boundary layers compared with the PFD scheme of uniform step.