How to adaptively resolve evolutionary singularities in differential equations with symmetry

Many time-dependent partial differential equations have solutions which evolve to have features with small length scales. Examples are blow-up singularities and interfaces. To compute such features accurately it is essential to use some form of adaptive method which resolves fine length and time scales without being prohibitively expensive to implement. In this paper we will describe an r-adaptive method (based on moving mesh partial differential equations) which moves mesh points into regions where the solution is developing singular behaviour. The method exploits natural symmetries which are often present in partial differential equations describing physical phenomena. These symmetries give an insight into the scalings (of solution, space and time) associated with a developing singularity, and guide the adaptive procedure. In this paper the theory behind these methods will be developed and then applied to a number of physical problems which have (blow-up type) singularities linked to symmetries of the underlying PDEs. The paper is meant to be a practical guide towards solving such problems adaptively and contains an example of a Matlab code for resolving the singular behaviour of the semi-linear heat equation.

许多含时偏微分方程的解会演变成具有小尺度特征的形式。例如爆破奇点和界面。要精确计算这些特征，必须使用某种形式的自适应方法，这种方法能够解析精细的长度和时间尺度，且实施起来不会过于昂贵。在本文中，我们将描述一种r -自适应方法（基于移动网格偏微分方程），它将网格点移动到解呈现奇异行为的区域。该方法利用了在描述物理现象的偏微分方程中经常出现的自然对称性。这些对称性有助于深入了解与发展中的奇点相关的（解、空间和时间的）尺度变换，并指导自适应过程。在本文中，将阐述这些方法背后的理论，然后将其应用于一些具有（爆破型）奇点且与潜在偏微分方程的对称性相关的物理问题。本文旨在成为自适应解决此类问题的实用指南，并包含一个用于解析半线性热方程奇异行为的Matlab代码示例。