Parallel Nonlinear Elimination Methods and Software for Partial Differential Equations

非线性偏微分方程（PDE）是广泛的重要应用领域的基本数学描述。特别是，这个项目将考虑在流体动力学，生物学和辐射扩散中出现的偏微分方程。由于它们的复杂性，这些方程只能用计算机数值求解，而且由于它们的特殊性质（冲击波、尖锐的锋面和局部奇点），即使在计算机上也很难求解。本计画将设计、分析及实作软体，以应用于一类迭代法数值求解非线性偏微分方程。该软件将以两种形式提供-- Matlab代码和与PETSc库互操作的软件包--供其他研究人员应用这些方法。技术上，该项目将研究求解具有不平衡非线性的代数非线性方程组的一类非线性消元算法。消除方法避免了传统方法的收敛速度慢时，局部奇异性出现识别“尺度不当”的非线性分量，并取代它们与其余更均匀的比例分量的函数。因此，设计的算法家族将获得并行域分解，可扩展性（相对于问题的大小），从多级方法，和鲁棒性（对局部奇点），从不完全消除。这些方法将在三个重要的应用类别上进行测试：跨音速可压缩流（CFD），心脏电波问题（计算生物学）和Marshak波问题（辐射传输）。所提出的算法和软件开发将对这三个应用产生巨大的影响，也将对需要求解大型非线性方程的计算科学的其他领域产生实质性的影响。

Nonlinear Partial Differential Equations (PDEs) are the basic mathematical description for a wide variety of important application areas. In particular, this project will consider PDEs that arise in fluid dynamics, biology, and radiation diffusion. Because of their complexity, these equations can only be solved numerically by computers, and because of their particular properties (shock waves, sharp fronts, and local singularities) they are difficult to solve even then. This project will design, analyze, and implement software for a class of iterative methods to numerically solve nonlinear PDEs. The software will be provided in two forms - Matlab codes and a package interoperating with the PETSc library - for other researchers to apply the methods.Technically, the project will study a class of nonlinear elimination algorithms for solving algebraic nonlinear equations with unbalanced nonlinearities. The elimination methods avoid traditional methods' slow convergence when local singularities appear by identifying "misscaled" nonlinear components and replacing them with a function of the remaining more uniformly scaled components. The family of algorithms thus devised will obtain parallelism from domain decomposition, scalability (with respect to problem size) from multilevel methods, and robustness (against local singularities) from incomplete elimination. The methods will be tested on three important classes of applications: transonic compressible flows (CFD), electric wave problems in the heart (computational biology), and Marshak wave problems (radiation transport). The proposed algorithm and software development will have a great impact on the three applications, and will also have substantial influence on other areas of computational science where large nonlinear equations need to be solved.