Multiscale numerical methods. I develop multiscale numerical methods---multi-implicit Picard integral deferred correction methods---for the integration of partial differential equations arising in physical systems with dynamics that involve two or more processes with widely-differing characteristic time scales (e.g., combustion, transport of air pollutants, etc.). These methods avoid the solution of nonlinear coupled equations, and allow processes to decoupled (like in operating-splitting methods) while generating arbitrarily high-order solutions.
Numerical methods for global atmospheric models. I have also been involved in the development and analysis of high-order numerical methods for weather prediction and climate modeling problems. I have developed numerical methods based on high-order splines and on double Fourier series in space, and combined these methods with a semi-Lagrangian semi-implicit time-stepping method. These methods were successfully tested using the shallow water equations, which have been used for decades by the atmospheric community as a testbed for promising numerical methods. I plan to apply the deferred correction approach to equations arising in global atmospheric models.