Parallel Nonlinear Elimination Methods and Software for Partial Differential Equations

非线性偏微分方程(PDE)是各种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(激波、锋面和局部奇异性)，即使在那时也很难求解。这个项目将设计、分析和实现一类迭代方法的软件来数值求解非线性偏微分方程组。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--供其他研究人员使用。在技术上，该项目将研究一类求解具有不平衡非线性的代数非线性方程的非线性消元算法。当局部奇异性出现时，消除方法避免了传统方法收敛速度慢的缺点，方法是识别“错位”的非线性分量，并用剩余的更均匀缩放的分量的函数来代替它们。这样设计的算法家族将从区域分解获得并行性，从多层方法获得可伸缩性(相对于问题大小)，以及从不完全消除获得稳健性(针对局部奇异性)。这些方法将在三个重要的应用类别上进行测试：跨音速可压缩流动(CFD)、心脏中的电波问题(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重大影响，也将对计算科学中其他需要求解大型非线性方程的领域产生实质性影响。

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.