Three Math Graduate Students have been awarded the Andrew Mellon Predoctoral Fellowships. These fellowships are awarded to students of exceptional promise and ability, who are enrolled or wish to enroll at the University of Pittsburgh in programs leading to the PhD in various fields of the humanities, the natural sciences and the social sciences.
Nan Jiang was awarded the Andrew Mellon Predoctoral Fellowship for work on developing a higher order simulation algorithm to perform ensemble calculation of Navier-Stokes euqations (NSE). There are many uncertainties inherent in numerical simulation of fluid flows. Calculation of an ensemble of solutions deals with these inherent uncertainties to increase the window of predictability and to quantify uncertainty. In this project an efficient, higher order, ensemble time discretization method incoporating an ensemble eddy viscosity model is being studied. This method results in linear systems that require significantly reduced computing efforts to advance in time.
Yong Li was awarded the Andrew Mellon Predoctoral Fellowship for work on mathematical analysis and computations ofpartial differential equations for transport and fluid flows of geophysics. In particular, I analyze the currently used time stepping schemes for weather and climate simulations, develop new algorithms with higher order accuracy, and implement the algorithms on realistic applications.
Pu Song was awarded the Andrew Mellon Predoctoral Fellowship for work on investigating numerical approximation of some coupled systems of partial differential equations in fluid mechanics and related problems. Coupling Stokes-Darcy model has been investigated actively in recent decades due to its broad application in real life such as predicting how contamination resulting from leaky underground storage tanks, chemical spills and various humanactives discharged into rivers, streams and lakes, flow in fractured porous media, blood flow in vessels and industrial filtration. Song's project is mainly focused on using efficient and accurate numerical methods to approximate the solution of this coupled model on irregular domains, which include deriving well-posedness of the scheme, error analysis and computational verification. The main challenge is the implementation of of the scheme on curved interfaces. Pu developed a method by using a multi-scale mortar multi-point flux mixed finite element method and a piecewise linear approximation to the curve to obtain optimal convergence. His ongoing work is on coupling Stokes-Darcy flows with transport on irregular domains. Implementation of a discontinuous Galerkin (DG) method with non-matching grids will be the main challenge for this project. He will also consider a stochastic model for this coupled problem such as stochastic source (rain fall) in fluid region and stochastic permeability in the porous media region. His final work will give an accurate simulation of the interaction between rain fall, river and underground water in a watershed near Pittsburgh.