Mathematical Sciences: Numerical Solution of Differential Equations in Mechanics

研究人员将开发和实施计算机模拟水波的算法，包括波浪破碎。这些算法将基于边界积分公式和时间推进算法来推进自由边界。将使用一些以前从未应用于水波问题的新的数值技术。特别是，将采用一种新的快速多层技术来求解积分方程组，这是求解过程的主要部分。研究人员将编写一个强大而高效的计算机程序，该程序将被记录下来，并放置在公共领域，供海洋学家、工程师、流体动力学家和数值分析师等不受限制地使用。研究人员将通过数值测试和数值分析彻底验证代码，并将其结果与实验室实验进行比较。这项工作将提高水动力学中计算机模拟的可靠性和效率，例如近海结构的模拟、船只在水中的运动和潮汐能发电。计算机算法的最新进展将首次应用于这些领域，使研究人员能够开发非常快的计算机硬件。通过开发一种公开可用的计算机代码，这项工作将有助于将思想从理论数学研究转化为技术应用。

The investigators will develop and implement algorithms for the computer modelling of water waves, including the breaking of waves. The algorithms will be based on a boundary integral formulation together with a time-marching algorithm to advance the free boundary. A number of new numerical techniques, never before applied to water wave problems, will be used. In particular, a new fast multilevel technique will be applied to solve the integral equations, which is the dominant part of the solution process. The investigators will write a robust and efficient computer program, which will be documented and placed in the public domain for unrestricted use, e.g., by oceanographers, engineers, fluid dynamicists, and numerical analysts. The investigators will validate the code thoroughly both through numerical testing and numerical analysis, and will compare its results with laboratory experiments. This work will improve the reliability and efficiency of computer simulations in hydrodynamics, such as arise in the modelling of off-shore structures, movement of boats through water, and tidal energy generation. Recent advances in computer algorithms will be adapted and applied for the first time in these areas, enabling the investigators to exploit very fast computer hardware. Through the development of a publicly available computer code this work will contribute to the transfer of ideas from theoretical mathematical research to technological applications.