An Optimal Time Stepping Method for Computational Science Applications

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

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 that can 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 impossible to solve efficiently and accurately using existing techniques, examples including the design of optimal drug structures in biochemistry, the study of 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.