标题： 超定射击方法用于以谱精度计算驻波 显示英文标题

标题： Overdetermined Shooting Methods for Computing Standing Water Waves with Spectral Accuracy

摘要： 一种高性能的射击算法被开发出来，用于以双精度和四精度在频域中计算自由表面欧拉方程的时间周期解，具有谱精度。该方法用于研究共振及其对驻波的影响。我们发现新的成核机制，在临界阈值以下的深度中，孤立的大振幅解以及此类解的闭合环突然存在。我们还研究了与行进情况下Wilton波相关的退化和次级分岔，并探讨了极端驻波波峰处自相似性的破坏。在浅水区域，我们发现驻波呈现为反复准弹性碰撞的反向传播孤立波。在有表面张力的深水中，我们发现驻波类似于反向传播的凹陷波。我们还讨论了解的存在性和非唯一性，以及傅里叶模态对波幅和流体深度的平滑与混乱依赖关系。在数值方法中，通过将问题表述为超定非线性系统，并使用基于伴随的最小化技术或二次收敛的信任区域方法来最小化目标函数，从而实现鲁棒性。通过在空间中使用谱配点法并可选地进行网格细化，在时间上使用高阶龙格-库塔法或谱延迟校正方法，并使用四精度以提高对参数空间精细区域的导航能力以及验证双精度结果，从而保持精度。简要讨论了GPU加速的实现问题，并针对多种硬件配置测试了该算法的性能。 显示英文摘要

摘要： A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton's ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counter-propagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth. In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high order Runge-Kutta or spectral deferred correction method in time, and quadruple-precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for GPU acceleration are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.