非交错的中心龙格库塔方法求解有干湿面的浅水方程组  

Unstaggered central Runge-Kutta method for shallow water equations with dry-wet fronts

作　　者：刘敏[1] 李订芳[1] 罗一鸣 董建 LIU Min;LI Dingfang;LUO Yiming;DONG Jian(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)

机构地区：[1]武汉大学数学与统计学院,湖北武汉430072

出　　处：《武汉大学学报（工学版）》2023年第2期148-159,225,共13页Engineering Journal of Wuhan University

基　　金：国家重点基础研究发展计划项目(编号:2017YFC0405901);国家自然科学基金项目(编号:51679143)。

摘　　要：提出一种具有和谐性、保正性和守恒性的非交错中心格式来求解带有干湿面的浅水方程组。根据间断底部的构造定义上、下水位阈值,再基于交错单元的水位值和水位阈值的大小关系判断非交错单元的干湿状态。对干界面和干湿面处的水位值,通过调整相邻单元的水位斜率使其得到修正,保证了格式的和谐性。采用中心龙格库塔方法和构造本质非振荡线性函数使格式在时间和空间方向均具有二阶精度。最后,通过数值算例验证了格式的和谐性、保正性、守恒性和稳健性。An unstaggered central scheme with well-balanced property, positivity preserving property and conservation, which can solve the shallow water equation with wet-dry fronts, is proposed in this paper. By defining upper and lower water level thresholds according to the structure of discontinuous bottom, the dry and wet states of unstaggered cells are judged based on the relationship between water level and water level threshold of staggered cells. By adjusting the water level slope of adjacent cells, the water level at dry fronts and dry-wet fronts are corrected to ensure the well-balanced property of the scheme. The scheme has achieved the second-order accuracy in time and space by using the central Runge-Kutta method and constructing the nonessentially oscillations linear function. Finally, the verification against benchmark tests shows that the scheme possesses well-balanced property, positivity preserving property, conservation and robustness.

关 键 词：浅水方程组 干湿面 非交错中心格式 和谐格式 

分 类 号：O241.82[理学—计算数学]