基于非均匀网格求解非线性对流扩散问题的一种高精度差分格式

A HIGH-ORDER FINITE DIFERENCE SCHEME FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS ON NONUNIFORM GRIDS

将作者基于均匀网格提出的优化差分法和反演差分法推广到非均匀网格中 ,提出了一种有效求解定常非线性对流扩散问题的高精度差分格式 ,在此基础上进一步发展了相应的非定常非线性对流扩散问题的高精度格式 .数值实例表明 ,该格式对对流占优和扩散占优问题均具有较好的适应性 ,对待求量的大梯度变化有极高的分辨能力 ,计算结果明显优于传统的差分格式 .此格式亦可方便地应用于非均匀网络在计算区域内取所有空间步长相等时的特例——均匀网络中 .在水环境模拟的实际计算中 ,根据待求量的变化规律合理地调整非均匀网络的疏密分布 ,不仅增强了高精度差分格式的实用效果 ,而且可使该格式获得比在含相同结点数的均匀网络系统中更为精确的数值结果 .

A high-order finite difference scheme using nonuniform grid s for steady nonlinear convection-diffusion problems is developed theoretically by the optimal difference method and the inverse difference method proposed by t he author and based on uniform grids. Subsequently a high-or der nonuniform grid scheme is developed for those unsteady cases. Compared with some conventional difference schemes, the present schemes have many advantages s uch as yielding more accurate numerical results, having high resolution for the large gradient changes of the unknown quantity, being well suitable for both convection-dominant flow and diffusion-dominant flow, and so on. The present schemes can also be conveniently used in uniform grids which are the special cas es of nonuniform grids in the case of the same grid steps being used within the computational domain. It is also pointed out that the appropriate structure of a nonuniform grid can not only make the present schemes more practical, but lead to a solution superior to that for a uniform grid structure with the same grid p oints in actual water environment simulation.