浅水方程源项处理的研究进展

许仁义 汪凌翔 王远见 方浩川 王俊贤 戴惠东

摘 要：浅水方程由纳维-斯托克斯方程推导而来，是一种描述具有自由表面的浅水体在重力作用下流动的数学模型。实际浅水流动不可能是完全理想的情况，必须考虑底坡和摩阻等源项的存在，它们影响计算的稳定性和精确性，如果处理不当还会造成格式的不和谐，因此源项处理是求解浅水流动方程的关键，也是国内外学者研究的一个重要方向。介绍目前求解浅水方程存在的4个主要困难，介绍浅水方程的各种离散方法以及优缺点，阐述源项处理的重要性和源项的分类，回顾和总结国内外处理源项的方法以及目前存在的困难。根据目前的研究进展，源项处理依旧有广阔的研究前景，如何平衡好源项处理方法的正效应和负效应是一个值得研究的课题。

关键词：浅水方程;源项处理;和谐性;有限体积法;不规则地形

中图分类号：TV131 文献标志码：A

doi：10.3969/j.issn.1000-1379.2021.07.007

引用格式：许仁义，汪凌翔，王远见，等.浅水方程源项处理的研究进展[J].人民黄河，2021，43（7）：35-40，83.

Abstract： The shallow water equation， derived from the Navier-Stokes equation is a mathematical model to describe the flow of shallow water with a free surface under the action of gravity. Actual shallow water flow cant be a perfectly ideal situation， bottom slope and friction must always be considered such as the existence of the source term. It affects the calculations stability and precision， if handles not properly it will cause disharmony in the format. Thus， the source term processing is the key to solve the shallow water flow equation and also is an important direction in the research of scholars at home and abroad. This paper introduced four main difficulties in solving shallow water equation and various discrete methods of shallow water equation as well as their advantages and disadvantages， expounded the importance of source term treatment and the classification of source term and reviewed and summarized the methods of source term treatment at home and abroad as well as the existing difficulties. Finally， according to the current research progress， it put forward that the source term treatment still had a broad research prospect and how to balance the positive and negative effects of source term treatment would be a worthy topic to be studied.

Key words： shallow water equation; source term processing; well-balance; finite volume method; irregular terrain

水力學中的浅水是指水深尺度远小于平面尺度且垂向流速小的水流[1]，人们关心的溃坝问题、水环境污染问题、潮汐和涌浪等都可以用浅水流动描述，而浅水方程是描述各类浅水流动的数学形式。如今，随着计算机性能大幅提升，浅水数值模拟以其无可替代的优势开始广泛应用于河道、河口、水库以及近海水域等环境。浅水方程是一种非线性双曲型方程，如何更加精确且高效地求解是一个值得研究的课题，因此衍生出许多离散方式。……