构造对流扩散方程高精度非均匀网格格式的指数变换方法

提出了一种基于非均匀差分网格,构造求解对流扩散方程的高精度格式的指数变换方法.引入指数函数,将对流扩散方程变换为扩散反应方程,消除了数值求解中较难处理的对流项.采用优化差分方法推导出扩散反应方程基于非均匀网格的高精度差分格式,进而通过逆变换得到对流扩散方程的高精度格式.理论分析表明,该方法具有3至4阶精度,当计算区域为均匀网格时取得4阶精度.数值实例表明,在相同的非均匀网格系统中,此方法的计算精度明显优于传统的隐式差分方法.在水环境的实际模拟计算中,根据物理量的变化规律灵活地调整非均匀网格的间距,不仅能增强高精度差分方法的实用性,而且可以取得比均匀网格方法更为精确的计算结果.

AN EXPONENTIAL TRANSFORMATION METHOD FOR CONSTRUCTING HIGH-ORDER SCHEMES OF CONVECTION-DIFFUSION EQUATIONS ON NON-UNIFORM DIFFERENCE GRIDS

An exponential transformation method for constructing high-order schemes to solve unsteady convection-diffusion equations(CDE) on non-uniform difference grids is developed.To eliminate the difficulty in dealing with convection terms,an exponential transformation function is introduced to turn the CDE into their equivalent diffusion-reaction equations(DRE).Through an extension of optimal difference method to non-uniform grids,the high-order compact scheme for the DRE is obtained,and then the corresponding scheme for the CDE is obtained through converse transformation.Theoretically,the proposed method has the third-to fourth-order accuracy;its fourth-order accuracy is achieved under uniform-grid settings.Two examples are provided to examine the performance of the proposed scheme.Compared with the traditional implicit difference scheme,the proposed scheme can give better results in the same non-uniform grid system.In the actual computation of water environment simulation,high-accurate methods using non-uniform grids with proper spacial steps would not only be more practical,but also produce more accurate numerical results than those using uniform grids,with the same number of grid points.