水波非线性演化分析的Hamilton理论及保辛算法研究

The evolution analysis of water waves plays a fundamental role in the structure design of ocean engineering. Analytical models and numerical methods constructed within Hamilton systems for the evolution of water waves can better preserve the physical properties of water waves, and can provide stable and precise numerical results for long time simulations. In this project, the Hamilton theory and the symplectic algorithm for the nonlinear evolution analysis of water waves will be studied using the Lagrange coordinate. The research contents are as follows: 1) Develop the Hamilton variational principle for the two-dimensional shallow water waves and establish the corresponding symplectic algorithm using the time finite element method. 2) Develop the constrained Hamilton variational principle for the two-dimensional water waves with arbitrary water depths, and construct the efficient and accurate symplectic methods for the linear and nonlinear evolution analyses of two-dimensional water waves with arbitrary water depths using the fast precise integration method and Zu Chongzhi-class method. 3) Based on the researches of 2), develop the constrained Hamilton variational principle for the three-dimensional water waves with arbitrary water depths, and establish the corresponding efficient and accurate symplectic methods. This project makes the best use of the merit that the Lagrange coordinate can satisfy moving boundaries accurately and the merit that numerical methods in Hamilton system are symplectic preserving. It will enrich the Hamilton theory and symplectic algorithm for water waves, push the numerical simulation for the nonlinear evolution analysis of water waves to the practical engineering applications, and hence have all-important academic value and wide application prospects.

水波演化分析是海洋工程结构设计的重要依据，在Hamilton体系下构建水波演化的近似模型和数值算法可以更好地保持水波问题的物理特性，长时间仿真数值稳定、精度好。本项目拟在Lagrange坐标下对水波演化的Hamilton理论和保辛算法进行研究：1）建立二维浅水波的Hamilton变分原理，采用时间有限元构造二维浅水波演化的保辛算法；2）建立二维任意水深水波的约束Hamilton变分原理，采用快速精细积分方法和祖冲之类算法，构造二维水波线性和非线性演化的高效精确算法。3）在2）的研究基础上，建立三维任意水深水波的约束Hamilton变分原理和高效精确保辛算法。本项目充分利用Lagrange坐标可精确处理动边界的优点以及Hamilton理论算法保辛的优点，对于发展水波问题的Hamilton理论和保辛算法，推动水波演化数值仿真在实际工程的应用，具有非常重要的理论价值和广阔的应用前景。

水波演化分析是海洋工程结构设计的重要依据，在Hamilton体系下构建水波演化的近似模型和数值算法可以更好地保持水波问题的物理特性，长时间仿真数值稳定、精度好。本项目在Lagrange坐标下对水波演化的Hamilton理论和保辛算法进行研究：1）研究了二/三维浅水、二/三维有限水深水波、完全非线性水波、二维内波、二维破碎波等各类水波问题的Hamilton理论，构造了这些水波问题的Lagrange作用量，所构造的作用量具有正定势能，可自动满足干湿面边界条件；2）基于有限元和离散Hamilton变分原理，构造出分析水波非线性演化分析的保辛算法的统一格式，所发展计算格式具有协调性，可保持静水稳定解，守恒性好，可长时间保持能量、动量等物理量，无质量损失，能保持干湿面处零水深，无虚假震荡。3）基于位移水波方程，给出了溃坝、孤立波、椭圆余弦波、机械激波等多种水波解析解，并给出这些波对应的位移，可模拟自然界中出现的极限涌波、波状涌潮等，本项目给出的孤立波速与实验波速吻合。4）将位移法扩展用于研究海洋内波，得到了浅水内孤立波和内机械激波的解析位移解，并用于分析海洋激流。本项目充分利用Lagrange坐标可精确处理动边界的优点以及Hamilton理论保辛的优点，对于发展水波问题的Hamilton理论和保辛算法，推动水波演化数值仿真在实际工程的应用，具有非常重要的理论价值和广阔的应用前景。