Prager Assistant Professor of Applied Mathematics

My research interests lie in numerical analysis and mathematical modeling, specifically in the following two areas: multiscale methods in materials science, particularly atomistic-to-continuum coupling methods for crystalline solids;computational electromagnetics, especially efficient energy-conserved splitting methods for the propagation of electromagnetic waves in large-scale field and for long-time duration.

Multiscale methods for crystalline materials

One of the most important goals of computational material science is to efficiently and reliably predict phenomena and to facilitate the design of new materials better able to resist failure. Intuitively, atomistic models should be the most accurate for this purpose. However, because of the immensely large number of atoms in a macroscopic material, it is not possible to only use atomistic models to simulate practical problems. One can use a continuum elasticity model to reduce the degrees of freedom. But, the continuum model is inaccurate near defects.Thus, atomistic-to-continuum coupling methods (a/c methods) have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects (the red region in the figure) with long-range elastic fields (the grey region in the figure).  The need for a more accurate a/c coupling method led me to develop several static consistent multiscale models for crystalline solids, and I have rigorously analyzed the stability and accuracy of them. The multiscale methods I have investigated so far are static methods at zero temperature and hence no dynamics are involved. However, to study equilibrium behavior at finite temperature, it is necessary to adopt a dynamical approach that allows the system to evolve in time. I want to investigate multiscale stochastic dynamical methods for crystalline simulation at finite temperature and study the accuracy of these methods.

## Energy-conserved splitting finite-difference time domain methods

The famous finite-difference time-domain (FDTD) method, first proposed by Yee in 1966, has been one important numerical algorithm for solving Maxwell’s equations in three-dimensional spaces. However, Yee’s scheme is only conditionally stable and constrained by the Courant-Friedrichs-Levy (CFL) condition on the time step. The alternating direction implicit (ADI)-FDTD methods successfully resolved the stability issue. They are unconditionally stable and have second order accuracy both in space and time. However, the ADI-FDTD schemes usually break the energy conservation in a lossless medium, an essential property of Maxwell’s equations. In collaboration with Wenbin Chen and Dong Liang, we have developed two new Energy-Conserved Splitting FDTD (EC-S-FDTD) algorithms. These methods retain not only all the properties of ADI-FDTD schemes, but also the energy conservation law in a lossless medium. In computational electromagnetics, the EC-S-FDTD method enables us to compute the long-time propagation of electric and magnetic waves efficiently and accurately.