Daniel R. Reynolds, SMU Mathematics -- Research

Brief introductory slides on my research (Fall 2008)

General Research Interests

My primary field of research is applied mathematics, specifically the sub-disciplines of scientific computation and numerical analysis. In the fast-developing field of scientific computation, there are three broad research thrusts that together aim to allow mathematical insight and innovation to make a difference in the physical, biological and engineering sciences: the incorporation of increased realism into mathematical modeling systems; the development of increasingly robust and accurate numerical methods for solution of mathematical models; and the development of computational algorithms to allow for efficient solution of these problems on increasingly-larger computational hardware.

In my own research, I have contributed to each of the aforementioned challenges through investigations on the algorithmic development and computational solution of coupled multi-physics systems of partial differential equations. These applications arise in real-world applications ranging from fusion energy to cosmological astrophysics and materials science. In these areas, I strive to explore three fundamental applied mathematics issues:

the accurate modeling of physical systems involving disparate time and space scales,
the development and use of highly-accurate and efficient time evolution algorithms for stiff multi-rate problems, and
the investigation of discretization and solution methods that retain constraint-preserving properties of PDE systems.

I pursue these investigations both through the general development of numerical methods for solving general multi-rate systems, as well as through a number of highly collaborative physics applications: the modeling of solid-state phase transformations and thermodynamics in shape memory alloys, simulations of macroscopic stability and refueling of fusion plasmas, calculations determining the physical processes underlying core-collapse supernova phenomena, and investigations of reionization physics within the early universe.