Hybrid asymptotic-numerical schemes for exponentially small selection mechanisms

[1]在流体力学和更广泛的物理科学中，有许多问题涉及在某种奇异极限下研究的非线性微分方程的渐近分析，其中潜在的选择机制决定了离散的可数无穷特征值序列。在某些具有挑战性的情况下，这种机制受到超出所有阶数的指数级小项的控制，由此产生的分析需要专门的技术。[2]发生这种情况的一个经典问题是在Saffman-Taylor粘性指法的上下文中，其中手指宽度的选择由表面张力参数中的指数小的项来确定。Saffman-Taylor问题的解决开创了指数渐近的现代方法。然而，这个问题包含了一些细微之处，使分析变得容易处理。特别地，前序表面无张力解是已知的闭合形式，这被证明是指数渐近应用的关键组成部分。控制粘性指进的类似选择机制有望适用于时间相关的流动或复杂几何形状的流动，但将渐近技术应用于这些扩展仍然是一个开放的挑战。[3]这篇论文将集中于渐近解或摄动解不能以封闭形式确定的问题，即使是在前导阶处也不能确定。在这种情况下，必须考虑将经典指数渐近方法扩展到混合格式，其中数值方法与分析理论结合使用。例如，这些数值方法涉及常微分方程解或偏微分方程解，以及随后对高维复值空间的实值解的解析延拓。然后，对解析延拓的研究产生奇点附近解的关键性质，然后将其编码到渐近格式中。[4]PHD将集中于这些解析和数值方法的发展，以及它们在连续介质和流体力学中的几个公开问题上的应用。应用将包括以下部分或全部：(I)研究二维喷嘴中的射流分离；(Ii)三维或轴对称喷嘴中的射流分离；以及(Iii)楔形中的Saffman-Taylor粘性指进。

[1] There are a number of problems in fluid mechanics and the wider physical sciences that involve the asymptotic analysis of nonlinear differential equations studied in some singular limit, where an underlying selection mechanism determines a sequence of discrete countably infinite eigenvalues. In certain challenging cases, this mechanism is governed by exponentially small terms beyond-all-orders and the resultant analysis demands specialised techniques. [2] One classic problem for which this occurs is in the context of Saffman-Taylor viscous fingering where the selection of the finger width is determined by terms exponentially small in the surface tension parameter. The resolution of the Saffman-Taylor problem pioneered modern methods in exponential asymptotics. However, the problem contains certain niceties that render the analysis tractable. In particular, the leading-order surface-tension-free solution is known in closed form and this turns out to be a crucial component in the application of exponential asymptotics. Similar selection mechanisms governing viscous fingering are expected to apply in generalisations to time-dependent flows or flows in complex geometries, but it remains an open challenge to adopt the asymptotic techniques to these extensions. [3] This thesis will focus on problems where the asymptotic or perturbative solutions cannot be determined in closed form, even at leading order. In such cases, it is essential to consider the extension of classical exponential asymptotic methodologies to hybrid schemes where numerical methods are used in conjunction with analytical theory. These numerical methods involve, for example, solutions of ordinary or partial differential equations and subsequent analytic continuation of real-valued solutions to higher-dimensional complex-valued spaces. The study of analytic continuation then yields key properties of solutions near singular points, which is then encoded into the asymptotic schemes.[4] The PhD will be focused on the development of these analytical and numerical methods, and their applications to several open problems in continuum and fluid mechanics. Applications will include some or all of the following: (i) the study of jet separation in a two-dimensional nozzle; (ii) jet separation in a three-dimensional or axi-symmetric nozzle; and (iii) Saffman-Taylor viscous fingering in a wedge.