An Optimal Time Stepping Method for Computational Science Applications

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.