浅水问题的约束Hamilton变分原理及祖冲之类保辛算法

The Constrained Hamilton Variational Principle for Shallow Water Problems and the Zu-Type Symplectic Algorithm

针对浅水流问题,将不可压缩条件作为约束处理,提出一种约束Hamilton变分原理,并利用该变分原理,推出一种基于位移和压强的浅水方程(SWE-DP).针对SWE-DP,构造了一种结合有限元和祖冲之类算法的混合数值方法.通过数值算例,将SWE-DP与两个现有的浅水方程进行了数值比较,从而验证了SWE-DP的可靠性,并验证了针对SWE-DP构造的数值算法的正确性.此外,数值算例还显示出祖冲之类算法在对浅水波进行长时间仿真时,具有很好的表现.

The shallow water problems were addressed.With the incompressible condition as the constraint,a constrained Hamilton variational principle was proposed for the shallow water problems.Based on the constrained Hamilton variational principle,the corresponding shallow water equations based on the displacement and pressure（SWE-DP） were developed.A hybrid numerical method combining the finite element method for the spatial discretization and the Zu-type symplectic method for the time integration was proposed to solve the SWE-DP.The correctness of the proposed SWE-DP is verified through the numerical comparisons of the present results with those from 2 sets of existing shallow water equations.The feasibility of the hybrid numerical method proposed for the SWE-DP is also proved through the numerical experiments.Moreover,the numerical experiments demonstrate the excellent performance of the Zu-type method for the simulation of the long time evolution of the shallow water motion.