An Optimal Time Stepping Method for Computational Science Applications

该方法首先利用高斯型结点和谱积分对时间方向进行离散化，然后利用延迟修正对耦合椭圆型方程进行预条件处理，每个修正过程只需要使用现有的快速椭圆方程解耦的解。然后利用迭代牛顿-克里洛夫技术有效地求解了预条件非线性系统。初步的数值实验表明，该方法是无条件稳定的，非常高效，在时间和空间上都可以达到任意阶的精度。特别是，没有观察到CFL约束，时间步长只取决于解的光滑度，因此是“最优的”。PI的初步结果亮点包括：(A)为大多数现有时间积分方案都无法解决的长时间电磁波模拟问题提供准确结果的时间域Maxwell方程求解器；以及(B)在大多数现有数值技术迅速崩溃的情况下保留具有奇异势的哈密顿系统结构的辛薛定谔方程求解器。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的Gandsfjorden的Sleipner A海上平台沉没，这是由于有限元分析不准确造成的，造成了近10亿美元的损失。这项建议的目的是利用先进的数学分析和开发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的数值技术。这个项目的成功将为科学和工程中的广泛应用带来新的工具和技术，这些应用是无法使用现有技术有效和准确地解决的，例如生物化学中的最佳药物结构的设计，天体物理学中的宇宙结构的研究，以及提高对地球系统科学中支配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，他们能够利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.