一种简单求解含源项浅水方程的高适用性方法  

A simple and widely adaptable numerical method for the shallow water equations with source terms

作　　者：艾欢 董建 陈炼钢[2] 李订芳[1] AI Huan;DONG Jian;CHEN Liangang;LI Dingfang(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China;State Key Laboratory of Hydrology,Water Resources and Hydraulic Engineering,Nanjing Hydraulic Research Institute,Nanjing 210029,China)

机构地区：[1]武汉大学数学与统计学院,湖北武汉430072 [2]南京水利科学研究院水文水资源与水利工程科学国家重点实验室,江苏南京210029

出　　处：《武汉大学学报（工学版）》2021年第3期189-196,共8页Engineering Journal of Wuhan University

基　　金：国家重点研发计划项目(编号:2016YFC0402207);国家自然科学基金项目(编号:51679143)。

摘　　要：针对浅水方程中对源项的处理方法较复杂且适用性低的问题,提出了一种计算简单的高适用性方法。采用合并源项与通量形式的浅水方程,避免对源项的复杂处理,方程中的新通量采用低耗散的Roe格式计算,并在通量分解时调整守恒变量差值,从而分解得到的波强包含了源项的影响,确保了格式的和谐性。此外,引入含通量限制器的反扩散项,在保持总变差不增(total variation diminishing,TVD)性质的同时提高格式精度,并根据Roe格式求解非线性方程时的特征分解性质,修改限制器参数来减小不必要的计算误差。通过5个经典算例验证了该方法的和谐性、高精度特性和高适用性。In consideration of the problem that the method dealing with source terms in the shallow water equations is complicated and has low applicability,a simple and widely adaptable numerical method is put forward.The shallow-water equation combining the source term and flux form is adopted to avoid the complicated processing of the source term.The low-dissipation Roe’s scheme is employed for the calculation of the new flux term in the equation,and the difference of conservation variables is adjusted during flux decomposition,so that the wave intensity obtained by decomposition includes the influence of the source term,and the harmony of the scheme is ensured.In addition,an anti-diffusion term with a flux limiter is introduced to improve the accuracy while maintaining the total variation diminishing(TVD)property,and then the parameters of the limiter are corrected according to the eigen-decomposition properties of solving nonlinear equations based on Roe’s scheme,so as to reduce unwanted calculation errors.The application in five benchmark tests shows that the proposed method is well-balanced with high accuracy and strong applicability.

关 键 词：复杂地形 和谐格式 二阶精度 Roe分解 浅水方程 

分 类 号：TV133[水利工程—水力学及河流动力学] O241.82[理学—计算数学]