Research

My research is focused on the development, analysis, and application of novel numerical methods for solving complex, high-dimensional differential equations arising from multiscale modeling and simulation. I am currently working on applying these methods to problems involving weather modeling, combustions, and plasmas.

The Polynomial Time Integration Framework

I am working to develop new time integration methods that can be derived by considering continuous interpolating polynomials in time. This approach generalizes classical polynomial-based methods like Adams-Bashforth, Adams-Moulton, and Backwards Differentiation Formula (BDF), and leads to new schemes with improved stability and accuracy. More generally, the polynomial framework greatly simplifies the construction of high-order general linear methods with varying degrees of parallelism and implicitness (i.e. explicit, semi-implicit, fully-implicit, and exponential methods).

Parallel-In-Time Methods for Non-Diffusive Equations

I am actively studying parallel-in-time (PinT) methods for solving non-diffusive equations. These types of equations present significant stability and convergence problems for many Pint methods, and remain and active area of research. My focus has been on analyzing and constructing stable and rapidly convergent Parareal methods for non-diffusive equations using Implicit-Explicit (IMEX) and exponential integrators.