Analysis of Fluid Motion

本文的目的是分析水波时变运动数值方法中的误差，设计改进的方法，并通过误差估计、收敛证明和计算试验来验证所提出方法的可靠性。感兴趣的方法称为边界积分法。它们在海洋工程和应用数学中得到了广泛应用。水面的运动由代表材料颗粒的标记来跟踪，这些标记是根据根据表面上的量计算的速度场来移动的。速度用奇异积分表示，因为控制方程是通过位势理论建立的。这一演化过程具有边界上的非线性、非局部波动特征。在这种方法中，数值不稳定性一直被观察到，但对二维情况的分析表明，通过适当的设计可以排除这些不稳定性。研究的第一个目标是发展稳定、收敛的数值方法，用于精确地模拟水平方向上周期性的、随时间变化的三维水波运动。因此，将处理完全的非线性运动，但将忽略与固体物体的相互作用。奇异积分的计算方法是一个主要问题，它可能与其他问题有关。第二个目标是处理固体物体与水面交汇处的边缘误差。流体和固体运动的实际预测通常是通过数值解表示运动的基本物理定律的方程来作出的。在海洋工程中，为了预报大水波的运动、水波与船舶或静止物体的相互作用以及波浪对墙壁或物体的作用力，已使用了这种方法。对这些问题和其他问题的计算遇到了数值不稳定性；也就是说，由于数值近似，小尺度可能会以一种与物理问题无关的方式增长。数学分析可以用来理解这种误差，并设计改进的方法。以前对二维波(那些在第三个方向上不变的波)的工作已经确定了这种特殊情况的误差来源。本工作的目的是在更困难和更现实的情况下进行这种分析，在这些情况下，错误的来源可能会更严重。希望数值方法的改进能够有助于可靠地预测水的波动。

DMS-9870091 J. Thomas Beale Abstract The purpose of the proposed work is to analyze the errors in numerical methods for the time-dependent motion of water waves, to design improved methods, and to verify the reliability of the proposed methods with error estimates, convergence proofs, and computational tests. The methods of interest are called boundary integral methods. They have been widely used in ocean engineering as well as in applied mathematics. The motion of the water surface is tracked by markers representing material particles which are moved according to a velocity field computed from quantities on the surface. The velocity is expressed in terms of singular integrals, since the governing equations are formulated through potential theory. The evolution has the character of nonlinear, nonlocal wave motion on the boundary. Numerical instabilities have long been observed in such methods, but analysis of the two-dimensional case has shown that they can be ruled out by proper design. The first goal of the proposed research is to develop stable, convergent numerical methods for the simulation of exact, time-dependent, three-dimensional water wave motion which is periodic in the horizontal directions. Thus the full nonlinear motion will be dealt with, but interactions with solid objects will be neglected. The method of calculation of the singular integrals is a primary issue, one which may have relevance to other problems. A second goal is to deal with the errors from edges where a solid object meets the water surface. Practical predictions of the motion of fluids and solids are often made by numerical solution of the equations which express the basic physical laws of the motion. Methods of the type studied here have been used in ocean engineering in order to predict the motion of large water waves, the interaction of water waves with ships or stationary objects, and the force exerted by a wave on a wall or object. Calculations of these probl ems and others have encountered numerical instabilities; that is, because of the numerical approximation, small scales may grow in a way unrelated to the physical problem. Mathematical analysis can be used to understand such errors and to design improved methods. Previous work for two-dimensional waves (those not varying in the third direction) has identified the sources of errors for this special case. The purpose of the present work is to carry out such analysis in more difficult and realistic cases where the sources of errors are likely to be more serious. It is hoped that improvements in the numerical methods could contribute to reliable predictions of water wave motion.