Hybrid asymptotic-numerical schemes for exponentially small selection mechanisms

指数小选择机制的混合渐近数值方案

• 批准号: 2427722
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2427722
• 关键词: Hybrid; asymptotic; numerical; schemes; exponentially

[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.

[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些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限特征值序列。在某些具有挑战性的情况下，这种机制是由指数小的条款超越所有的订单和由此产生的分析需要专门的技术。[2]一个经典的问题，这发生在萨夫曼-泰勒粘性指进的情况下，其中的手指宽度的选择是由表面张力参数的指数小的条款确定。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节，使得分析变得容易处理。特别是，领先的顺序表面张力自由的解决方案是已知的封闭形式，这原来是一个关键组成部分，在指数渐近的应用。类似的粘性指进的选择机制，预计适用于在一般情况下，以复杂的几何形状的时间相关的流动或流动，但它仍然是一个开放的挑战，采用渐近技术，这些扩展。[3]本论文主要研究的问题是渐近解或摄动解不能以封闭形式确定，即使是在首阶。在这种情况下，它是必不可少的，以考虑扩展的经典指数渐近方法的混合计划，其中数值方法与分析理论结合使用。这些数值方法涉及，例如，解决方案的普通或偏微分方程和随后的分析延续的实值解决方案，以更高的维复值空间。解析延拓的研究，然后产生奇异点附近的解决方案，然后编码到渐近计划的关键属性。[4]博士将专注于这些分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开放问题的应用。应用将包括以下部分或全部：（i）二维喷嘴中射流分离的研究;（ii）三维或轴对称喷嘴中的射流分离;以及（iii）楔形体中的Saffman-Taylor粘性指进。