四点时空偏心隐格式的改进求解

Two improved calculation methods of Preissmann four-point linear implicit scheme

在传统Preissmann格式离散求解一维浅水方程的基础上,提出两个四点时空偏心隐格式改进算法.一个是使用相邻计算单元在同一点上流量守恒的关系来建立追赶法矩阵求解;二是在相邻单元的基础上再取一个和两单元都相交的控制体,对其使用连续方程并进行有限体积离散的方法,同样建立了追赶法矩阵求解.求解过程中,这两个方法都创新地把流量和水深进行了解耦,构造成只有一个变量的追赶法矩阵.通过理论解及传统解法验证了这两个新方法,得到了较好的结果.新方法继承了Preissmann格式的优点,拓展了Preissmann格式的求解思路.

Based on the Preissmann four-point linear implicit finite-difference scheme, two new improved calculation methods are analyzed to resolve the one-dimension shallow-water equations. Firstly, based on the conservation of flux at the point between two adjacent control volumes, the governing equations discretized by Preissmann scheme are reconstructed by a new way. Secondly, the continuum equation is discretized by a classical finite difference method in a new control volume which includes a part of two adjacent control volumes. Based on the relationship built in the new control volume, the governing equations discretized by Preissmann scheme are reconstructed too. The two improved calculation methods both can make the flux and flow depth or their varieties of amount be calculated respectively. Moreover, these equations reconstructed by these two improved methods, whose coefficients are tridiagonal matrixes, all can be resolved by the method of forward elimination and backward substitution. These new methods are well confirmed by theoretical and classical results.