Space-Time Conservation Method Applied to Saint Venant Equations
Space-Time Conservation Method Applied to Saint Venant Equations
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DOI：
10.1061/(asce)0733-9429(1998)124:5(501
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发表时间：
1998-05
影响因子：
2.4
通讯作者：
Thomas Molls;F. Molls
中科院分区：
文献类型：
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作者：
Thomas Molls;F. Molls
本文描述了Chang提出的一种新的数值方法，并将其用于求解一维（1D）g和二维（2D） Saint Venant方程。这种新方法不同于传统的数值方法（即有限差分法、有限元法、有限体积法、光谱法等）。Chang的方法将空间和时间放在同一个基础上，因此空间和时间是统一的，这是新方法区别于其他技术的关键特征。这种方法是明确的，使用交错网格，在空间和时间上强制通量守恒，并且不需要上旋，通量分裂，通量限制器，特征值的评估或人工粘度的添加。此外，该方案简单，易于实现（见附录1），并且可以扩展到更高的维度。首先，介绍、解释了由Chang开发的新技术，并将其应用于1D Saint Venant方程。为了说明它的有效性，模拟了一个理想的溃坝和直矩形河道中的水力跳跃。用新方法得到的数值结果与其他类似复杂性的常规方法的结果、实验数据和解析解进行了比较。其次，将Chang的格式推广到二维Saint Venant（即浅水或深度平均）方程的解中。作者没有遵循Chang的二维发展，而是采用了Strang的分步方法，将二维问题转化为两个一维问题。将新的二维格式应用于斜跃式水力跳跃，并与解析解进行了比较。
A new numerical technique by Chang is described and used to solve the one-dimensional (1D)g and two-dimensional (2D) Saint Venant equations. This new technique differs from traditional numerical methods (i.e., finite-difference, finite-element, finite-volume, spectral methods, etc.). Chang’s method treats space and time on the same footing, so that space and time are unified—key characteristic that distinguishes the new method from other techniques. This method is explicit, uses a staggered grid, enforces flux conservation in space and time, and does not require upwinding, flux-splitting, flux limiters, evaluation of eigenvalues, or the addition of artificial viscosity. Furthermore, the scheme is simple, easy to implement (see Appendix I), and can be extended to higher dimensions. First, the new technique, as developed by Chang, is introduced, explained, and applied to the 1D Saint Venant equations. To illustrate its effectiveness, an idealized dam-break and a hydraulic jump in a straight rectangular channel are simulated. The numerical results obtained using the new method are compared with results from other, more conventional techniques of similar complexity, experimental data, and an analytical solution. Next, Chang’s scheme is extended for solution of the 2D Saint Venant (i.e., shallow water or depth-averaged) equations. The writers do not follow Chang’s 2D development but instead employ Strang’s method of fractional steps, which transforms the 2D problem into two 1D problems. The new 2D scheme is applied to an oblique hydraulic jump, and the results are compared with an analytical solution.