1. Strong stability preserving time stepping methods for hyperbolic PDEs
2. Positivity preserving and asymptotic preserving time evolution methods for kinetic equations with hyperbolic limits
3. Error inhibiting time-discretizations
4. Developing energy and power-efficient numerical methods
5. Weighted essentially non-oscillatory methods for problems with sharp gradients and shocks
6. Reduced order methods in a collocation setting
7. Spectral collocation methods
My research interests lie primarily in the development of numerical algorithms for the simulation of hyperbolic PDEs. I am best known for my research in strong stability preserving (SSP) time discretizations. These comprise many types of time-evolution methods including Runge–Kutta methods, GLMs, multi-derivative methods, and integrating factor methods. Such methods can be explicit or implicit, and semi-implicit (IMEX) extensions have also been useful. I am also interested in spatial discretization techniques such as spectral and pseudospectral methods and WENO methods. More recently, I have been developing reduced basis methods in a collocation context.