Semi-implicit finite difference methods for the two-dimensional shallow water equation
Semi-implicit finite difference methods for the two-dimensional shallow water equation
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DOI：
10.1016/0021-9991(90)90091-e
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发表时间：
1990-01
影响因子：
4.1
通讯作者：
V. Casulli
中科院分区：
文献类型：
--
作者：
V. Casulli
本文导出并讨论了二维浅水波方程的半隐式有限差分方法。首先对控制方程进行特征分析，确定隐式离散的项，使方法的稳定性不依赖于坐标。这些项是动量方程中的水面高程梯度和连续性方程中的速度发散。对流项显式离散。对流项的显式离散是迎风离散，它是条件稳定的，并引入了一些人工粘性。结果表明，当采用大时间步长的欧拉-拉格朗日方法离散对流项时，消除了稳定性限制，减少了人为粘性。该方法在每个时间步都需要求解一个线性、对称、5对角系统。这样的系统是对角占优的，正元素在主对角线上，负元素在其他地方。从而保证了数值解的存在唯一性。由此产生的算法是质量保守的，完全矢量化的现代矢量计算机上的有效实施。当与ADI技术结合使用时，该方法的性能得到进一步改善，ADI技术产生两组更简单的线性3对角系统，并保持上述所有特性。
In this paper a semi-implicit finite difference method for the 2-dimensional shallow water equations is derived and discussed. A characteristic analysis of the governing equations is carried out first, in order to determine those terms to be discretized implicitly so that the stability of the method will not depend upon the celerity. Such terms are the gradient of the water surface elevation in the momentum equations and the velocity divergence in the continuity equation. The convective terms are discretized explicitly. The simpler explicit discretization for the convective terms is the upwind discretization which is conditionally stable and introduces some artificial viscosity. It is shown that the stability restriction is eliminated and the artificial viscosity is reduced when an Eulerian-Lagrangian approach with large time steps is used to discretize the convective terms. This method, at each time step, requires the solution of a linear, symmetric, 5-diagonal system. Such a system is diagonally dominant with positive elements on the main diagonal and negative ones elsewhere. Thus, existence and uniqueness of the numerical solution is assured. The resulting algorithm is mass conservative and fully vectorizable for an efficient implementation on modern vector computers. The performance of this method is further improved when used in combination with an ADI technique which results in two sets of simpler, linear 3-diagonal systems and maintains all the properties described above.