An Optimal Time Stepping Method for Computational Science Applications

这个建议的重点是对一类新的Krylov延迟校正加速的“线转置法”的数学分析和有效的实施与时间相关的偏微分方程。该方法首先离散的时间方向使用高斯型节点和谱积分，和由此产生的耦合椭圆方程的预处理使用延迟校正，其中每个校正过程只需要一个解耦系统的解决方案，使用可用的快速椭圆方程求解器。预处理的非线性系统，然后有效地解决使用迭代牛顿-克雷洛夫技术。初步的数值实验表明，该方法是无条件稳定的，非常有效，并可以达到任意数量级的精度在时间和空间。特别是，没有CFL的约束条件已被观察到，时间步长仅取决于平滑的解决方案，因此是“最佳的”。 PI的初步结果的亮点包括：（a）时域麦克斯韦方程求解器，它为大多数现有的时间积分方案失败的长时间电磁波模拟问题提供了准确的结果;和（B）辛薛定谔方程求解器，它保留了具有奇异势的哈密顿系统的结构，而大多数现有的数值技术很快就崩溃了。众所周知，不准确的数值算法导致了许多代价高昂的项目失败，例如1991年8月23日挪威斯塔万格附近Gandsfjorden的Sleipner A海上平台的沉没，这是由于不准确的有限元分析，造成了近10亿美元的损失。本课题的目的是利用先进的数学分析方法和数值技术，为重要的科学和工程问题提供准确、稳定的数值模拟结果。特别是，PI将研究和实现一类新的数值算法，用于解决由偏微分方程建模的时间依赖问题。该项目的成功将为科学和工程领域的广泛应用带来新的工具和技术，这些应用是使用现有技术无法有效和准确解决的，例如生物化学中最佳药物结构的设计，天体物理学中的宇宙结构研究，并提高了对地球系统科学中水文过程的物理学的理解。该项目还侧重于培训新一代科学家，能够使用复杂的数学理论开发先进的数值工具。

The focus of this proposal is on the mathematical analysis and efficient implementation of a new class of Krylov deferred correction accelerated "method of lines transpose" for time dependent PDE's. The method first discretizes the temporal direction using Gaussian type nodes and spectral integration, and the resulting coupled elliptic equations are preconditioned using deferred corrections, in which each correction procedure only requires the solution of a decoupled system using available fast elliptic equation solvers. The preconditioned nonlinear system is then solved efficiently using iterative Newton-Krylov techniques. Preliminary numerical experiments show that this method is unconditionally stable, very efficient, and can achieve arbitrary order of accuracy in both time and space. In particular, no CFL constraints have been observed and the time step size only depends on the smoothness of the solution and hence is "optimal". Highlights of the PI's preliminary results include (a) a time domain Maxwell equation solver which provides accurate results for a long-time electromagnetic wave simulation problem for which most existing time integration schemes fail; and (b) a symplectic Schrodinger equation solver which preserves the structure of a Hamiltonian system with singular potential while most existing numerical techniques quickly blow up. It is well known that inaccurate numerical algorithms have caused many costly project failures, examples include the sinking of the Sleipner A offshore platform in Gandsfjorden near Stavanger, Norway, on August 23, 1991, which was due to inaccurate finite element analysis and resulted in a loss of nearly one billion dollars. The purpose of this proposal is to use advanced mathematical analysis and develop numerical techniques thatcan efficiently provide accurate and stable numerical simulation results to important science and engineering problems. In particular, the PI will study and implement a novel class of numerical algorithms for time dependent problems modeled by partial differential equations.The success of this project will bring new tools and techniques to a wide class of applications in science and engineering that are impossibleto solve efficiently and accurately using existing techniques, examples including the design of optimal drug structures in biochemistry, the studyof cosmos structure in astrophysics, and improved understanding of the physics that govern hydrologic processes in Earth system science. This project also focuses on the training of a new generation of scientists capable of developing advanced numerical tools using sophisticated mathematical theory.