非线性预条件算法及其应用研究

Nonlinear preconditioning is a globalization technique for improving global convergence properties of Newton's method applied to systems of equations with unbalanced nonlinearities, and the key idea is to replace nonlinear functions or change unknowns. In the numerical simulation, we often solve large systems of equations arising from discretization of nonlinear partial differential equations (PDEs) , therefore, it plays an important role in applications. In this project, we study the design and applications of nonlinearly preconditioned parallel algorithms based on PETSc (Portable, Extensible Toolkit for Scientific Computation): (1) for transonic full potential flow around airfoils, we design a parallel algorithm by combining the Newton-Krylov method, the Schwarz method as preconditioners and nonlinear preconditioning techniques. In the transonic case, the algorithm accelerates convergence by removing local strong nonlinearities near the shock, which differs from the traditional Newton-like algorithm suffering from a lengthy residual stagnation period. (2) there are no any results reported about applications of nonlinear preconditioning in combustion, so we plan to apply nonlinear preconditioning techniques with physics-based field-split partitioning to solve the flame sheet model with strong nonlinearities, coupling the Navier-Stokes equation with a Shvab-Zeldovich formulation, so that the number of nonlinear iterations can be dramatically reduced.

非线性预条件是一种改善求解强非线性问题牛顿法收敛性质的全局化技术，其核心思想是替换非线性函数或改变未知量。在数值模拟中，我们常常要求解离散后得到的大规模非线性系统，因而该技术有着重要的应用价值。基于科学计算工具箱PETSc，本项目拟研究非线性预条件并行算法的设计及其应用：(1)针对机翼绕流跨声速全速势模型，本项目结合Newton-Krylov算法、Schwarz预条件和非线性预条件等技术，构造并行的核心求解算法。在跨声速的情况时，该算法通过移除激波附近局部强非线性的方式来加速收敛，而不需要像传统的牛顿类算法经历冗长的残量停滞期；(2)目前没有文献报告非线性预条件技术在燃烧领域的研究成果，本项目拟研究如何利用基于物理场分裂的非线性预条件技术，求解由Navier-Stokes方程和Shvab-Zeldovich方程耦合的具有强非线性的火焰面模型问题，以达到显著减少牛顿迭代数的目的。

非线性预条件技术能够有效改善强非线性问题的全局收敛性质并且显著减少牛顿迭代数，因而在科学和工程应用中具有重要的价值。本项目以空气动力学和燃烧模型等实际应用为背景，开展了非线性预条件并行算法的设计、改进及其应用等方面的研究:（1）针对具有局部强非线性的问题，以跨音速全速势模型为例，提出一种全新的非线性预条件并行算法，数值实验表明该算法可以有效减少外部牛顿迭代数，从而加速收敛。（2）对于跨音速全速势方程，提出一种改进的挑选被消去变量的策略，基于该策略构造的右侧非线性预条件并行求解器，在计算代价和计算时间方面优于现有的基于局部马赫数消去的ANE算法。（3）对于火焰面模型，以构造有效消除强非线性的非线性预条件为目的，在物理子问题的合理分组和求解顺序等方面开展研究，并且通过火焰面模型、自然对流-顶盖驱动方腔流等数值例子，验证了所给出的INB-NE和MSPIN两种非线性预条件算法近似误差界估计的有效性。部分结果发表在《SIAM Journal on Scientific Computing》、《Journal of Computational Physics》等国际权威期刊上。