An Optimal Time Stepping Method for Computational Science Applications

本文的重点是对一类新的Krylov延迟校正加速“线转置法”的数学分析和有效实现。该方法首先利用高斯型节点和谱积分对时间方向进行离散化，得到的耦合椭圆方程采用延迟修正进行预处理，每次修正过程只需要使用可用的快速椭圆方程解算器解耦。然后利用迭代牛顿-克雷洛夫技术有效地求解了预条件非线性系统。初步数值实验表明，该方法具有无条件稳定、高效的特点，可以在时间和空间上实现任意阶的精度。特别是，没有观察到CFL约束，时间步长仅取决于解决方案的平滑性，因此是“最优”的。PI初步结果的亮点包括：(a)时域Maxwell方程求解器，它为大多数现有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；(b)一个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失效了。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，其中包括1991年8月23日挪威斯塔万格附近的gandsjorden的Sleipner A海上平台沉没，这是由于不准确的有限元分析造成的，造成了近10亿美元的损失。本提案的目的是利用先进的数学分析和发展数值技术，能够有效地为重要的科学和工程问题提供准确和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，用于由偏微分方程建模的时间相关问题。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，这些应用是现有技术无法有效和准确地解决的，例如生物化学中最佳药物结构的设计，天体物理学中宇宙结构的研究，以及提高对地球系统科学中控制水文过程的物理学的理解。该项目还侧重于培养能够利用复杂的数学理论开发先进数值工具的新一代科学家。

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.