Xingjie Helen Li 

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 atomistictocontinuum coupling methods for crystalline solids;computational electromagnetics, especially efficient energyconserved splitting methods for the propagation of electromagnetic waves in largescale field and for longtime duration.
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, atomistictocontinuum 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 longrange 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. Energyconserved splitting finitedifference time domain methodsThe famous finitedifference timedomain (FDTD) method, first proposed by Yee in 1966, has been one important numerical algorithm for solving Maxwell’s equations in threedimensional spaces. However, Yee’s scheme is only conditionally stable and constrained by the CourantFriedrichsLevy (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 ADIFDTD 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 EnergyConserved Splitting FDTD (ECSFDTD) algorithms. These methods retain not only all the properties of ADIFDTD schemes, but also the energy conservation law in a lossless medium. In computational electromagnetics, the ECSFDTD method enables us to compute the longtime propagation of electric and magnetic waves efficiently and accurately. 
