AF: Small: Fully Implicit Methods for Partial Differential Equations and Software for Hybrid Architecture

该项目的第一个目标是开发新的区域分解算法和软件，用于多物理场应用中产生的一些高度非线性，耦合的偏微分方程系统的数值解。对于简单的问题，隐式方法相对容易从给定的显式或半隐式方法发展出来，但对于一些多物理场问题，开发一种允许高性能实现的完全隐式方法是相当困难的。现有的技术大多是非光滑的，因此难以用牛顿型求解器求解。在本项目中，将发展一些离散化技术，包括用于捕获域信息的高阶非光滑离散化和用于构建牛顿迭代的雅可比系统的低阶局部光滑离散化。低阶离散化作为非线性预条件，加快了收敛速度，但不改变解的精度。该项目的第二个目标是在具有大量处理器的高性能计算机上开发所提出的线性和非线性预处理方法的有效实现。对于许多类型的单物理问题，包括算法和软件在内的相对成熟的技术是可以解决的，但对于耦合的多物理问题，迫切需要鲁棒和可扩展的技术，特别是对于带有加速器的大型并行计算机。所提出的算法和软件将对几个重要的应用领域产生重大影响，例如全球大气流动和生物流体的模拟，并将对需要求解大型线性和非线性方程的计算科学的其他领域产生重大影响。为了扩大研究的影响，该软件将与广泛使用的PETSc软件包完全兼容。对于对高性能计算和一般计算科学与工程感兴趣的研究生和本科生来说，该研究是一个机会丰富的领域。

The first goal of the project is to develop new domain decomposition algorithms and software for the numerical solution of some highly nonlinear, coupled systems of partial differential equations arising from multi-physics applications. For simple problems, implicit methods are relatively easy to develop from a given explicit or semi-implicit method, but for some multi-physics problems it is quite difficult to develop a fully implicit method that allows a high performance implementation. Most of the existing techniques are non-smooth and therefore difficult to solve with Newton type solver. In the project, some discretization techniques will be developed that involve high order non-smooth discretization for capturing the domain information, and low order locally smooth discretization for building the Jacobian system of the Newton iterations. The low order discretization serves as a nonlinear preconditioner that speeds up the convergence, but doesn't change the accuracy of the solution. The second goal of the project is to develop an efficient implementation of the proposed linear and nonlinear preconditioning approaches on high performance computers with a large number of processors. Relatively mature technologies including algorithms and software are available for solving many types of single physics problems, but for coupled multi-physics problems, robust and scalable techniques are badly needed, especially for large scale parallel computers with accelerators. The proposed algorithms and software will have a great impact on several important application areas, such as the simulation of global atmospheric flows and the bio-fluids, and will also have substantial influence on other areas of computational sciences where large linear and nonlinear equations need to be solved. To broaden the impact of the research, the software will be made fully compatible with the widely used PETSc package. The research is a rich area in opportunities for both graduate and undergraduate students interested in high performance computing and general Computational Science and Engineering.