A Practical Approach to Rothe's Method: Method of Lines Transpose

物理、化学和材料科学中的许多重要方程都显示出一系列关键的时间尺度。通常，行为分为快、中、慢三种时间尺度。以浇注聚合物膜的工艺为例，该聚合物膜在浇铸过程中经历相分离。这类聚合物包括燃料电池膜、现代电池中的隔膜和基于聚合物的太阳能电池。最快的时间尺度是旋转节点区(铸造过程中的初始过程)，可能是微微秒到微秒。然后是一个更长的瞬变阶段，大约是几十分钟到几个小时。这个瞬变是当系统稳定下来，膜开始形成其最终形式时。整个铸造过程可能需要几天时间才能完成，这决定了薄膜的工作状态。对这些过程建模的最新技术状态是通过使用分子动力学对系统中的单个原子进行建模，使用直接模拟铸造。然而，即使在最大的超级计算机上，最好的方法也只能模拟数百皮秒。为了在这些时间尺度上获得这些系统的精确模型，提出了一类新的模型，功能化的Cahn Hilliard。然而，这类模型的求解非常具有挑战性，需要在大范围的尺度上同时具有时间和空间精度。为了适应这个模型，我们正在开发一类新的数值方法，这些方法利用了多核计算革命的优势。如果成功，这类新的数值方法将促进对我们过去只能通过实验询问的问题的快速模拟。开发这类新的数值方法来挑战这种性质的模型的总体目标是将设计过程从爱迪生式的方法转移到深思熟虑的设计过程中。有了可靠的数字工具，设计过程可以大大提高。这方面的一个关键例子是新型波音777飞机的材料和机翼设计，它主要是通过计算机模拟进行设计的。这项提议的核心是为一大类偏微分方程组开发O(N)、半解析、高阶、隐式求解器，该方法基于直线转置方法，也称为Rothe方法。这些方法的动机是PI致力于发展一种对所有阶的A-稳定的时间隐式方法来求解可变波速的声学问题。该方法首先对偏微分方程组进行时间离散化，然后用快速求和法求解得到的非振动Helmholtz方程，即用自由空间格林函数求逆算子，然后用边界积分修正自由空间解。为了产生高阶解，提出了一种基于逐次卷积的新方法。该提议的核心是将基于逐次卷积的核心算法扩展到更广泛的线性和非线性偏微分方程组。提出了一种新的多层区域分解方法。DD方法为为分布式多核平台开发算法的可扩展版本提供了一条可能的途径。

Many important equations in Physics, Chemistry and Materials Science exhibit a range of critical time scales. Generally, behaviors break up into a range of fast, medium, and slow time scales. Take for example the process of casting a polymer membrane that undergoes phase separation during the casting process. Polymers of this type include fuel cell membranes, separators in modern batteries, and polymer based solar cells. The fastest time scale is the spinodal region (an initial coursing process during casting) which might be pico to micro seconds. Then comes a longer transient phase, which is on the order of tens of minutes to hours. This transient is when the system settles down and the membrane starts to take its final form. The overall casting process can take days to complete, dictating how well the membrane will work. The state of the art for modeling these processes is to use direct simulation of casting by modeling the individual atoms in the system using molecular dynamics. However, even on the biggest super computers, the best methods can only simulate hundreds of pico seconds. To obtain accurate models of these systems over these time scales, a new class of models, functionalized Cahn Hilliard, was proposed. However, this type of model is very challenging to solve, requiring both temporal and spatial accuracy over a wide range of scales. To accommodate this model, we are developing a new class of numerical methods which take advantage of the multi-core computing revolution. If successful, this new class of numerical methods will facilitate rapid simulations of problems we could only experimentally interrogate in the past. The overall goal in developing this new class of numerical methods for challenging models of this nature is to move the process of design away from an Edisonian approach to one of thoughtful design process. With reliable numerical tools, the process of design can be greatly enhanced. A key example of this is the materials and wing design of the new Boeing 777 aircraft, which was designed primarily through computer simulation.This proposal centers on the development of O(N), semi-analytic, high order, implicit solvers based on the method of lines transpose, otherwise known as Rothe's method, for a large class of PDEs. The methods are motivated by the PIs work on developing an A-Stable to all orders in time implicit method for acoustic problems with a variable wave speed. The method starts by discretizing the PDE in time, then solving the resulting non-oscillatory Helmholtz equation using a fast summation methodology, i.e., we use the free space Green's function to invert the operators followed by applying a boundary integral to correct the free space solution. To generate high order solutions, a new approach based on successive convolution is introduced. The proposal centers on the extension of the core algorithm, based on successive convolution, to a wide class of linear and non-linear PDEs. A novel method for multi-level domain decomposition (DD) is presented. The DD method offers a possible path for developing scalable versions of the algorithm for distributed multi-core platforms.