Todd Dupont Professor, Departments of Computer Science and Mathematics and the James Franck Institute Senior Fellow, Computation Institute
My research deals with the analysis, evaluation and construction of numerical methods to approximate the solutions of partial differential equations (PDEs).
The question of how to make effective use of computers with multiple processing units is one that is being investigated in several ways. I have recently produced several schemes that involve decomposing the computational domain into subregions and organizing the computation so that the work on each of these subdomains can be done almost independently of the others. This work was for parabolic PDEs and I am studying its extension.
Including adaptivity in numerical methods can make them more robust and efficient. Most simulations of time dependent problems use adaptivity for the control of the time step, and substantial progress has been made by many people in understanding how to control the spatial mesh when approximating PDEs. I have worked on this for several years.
I am currently collaborating with physicists and mathematicians on questions related to instabilities and singularity development in the flow of fluids and psuedo fluids.