标题： 半显式解及其在非线性哈密顿耦合模态理论中的作用 显示英文标题

标题： Semi-explicit solutions to the water-wave dispersion relation and their role in the nonlinear Hamiltonian coupled-mode theory

摘要： 哈密顿耦合模式理论（HCMT），最近由Athanassoulis和Papoutsellis [1]推导出来，为在任意地形上求解完全非线性水波问题提供了一种高效的新方法。 在HCMT中，大量使用了一个局部的、具有变化参数的水波色散关系的根，这些根必须在每个水平位置和每个时间瞬间进行计算。 因此，这些根的快速且精确的计算，适用于所有可能的变化参数值，具有根本的重要性。 本文推导出新的、半显式且高度精确的根查找公式，特别是针对衰减模式对应的根。 推导基于逐次应用一种Picard型迭代和Householder根查找方法。 得到了精度非常好的显式近似公式，这些公式足以支持HCMT在多种应用中的使用。 在大多数要求苛刻的情况下，例如非常陡峭的深水波，仅需三次迭代即可使用显式形式作为初始值，实现所需根的机器精度确定。 在HCMT中利用这种根查找过程，得到一个高效的数值求解器，能够处理任意地形上的完全非线性水波。 对要求苛刻的非线性问题的应用证明了该方法的效率和鲁棒性。 显示英文摘要

摘要： The Hamiltonian coupled-mode theory (HCMT), recently derived by Athanassoulis and Papoutsellis [1], provides an efficient new approach for solving fully nonlinear water-wave problems over arbitrary bathymetry. In HCMT, heavy use is made of the roots of a local, water-wave dispersion relation with varying parameter, which have to be calculated at every horizontal position and every time instant. Thus, fast and accurate calculation of these roots, valid for all possible values of the varying parameter, are of fundamental importance. In this paper, new, semi-explicit and highly accurate root-finding formulae are derived, especially for the roots corresponding to evanescent modes. The derivation is based on the successive application of a Picard-type iteration and the Householder's root finding method. Explicit approximate formulae of very good accuracy are obtained, which are adequate to support HCMT for many types of applications. In most demanding cases, e.g. very steep, deep-water waves, machine-accurate determination of the required roots is achieved by no more than three iterations, using the explicit forms as initial values. Exploiting this root-finding procedure in the HCMT, results in an efficient, numerical solver able to treat fully nonlinear water waves over arbitrary bathymetry. Applications to demanding nonlinear problems demonstrate the efficiency and the robustness of the present approach.