A first order accurate in time, finite difference scheme is proposed and analyzed for the Ericksen-Leslie system, which describes motions of nematic liquid crystals. For the penalty function to approximate the phase field constraint, a convex-concave decomposition for the corresponding energy  functional is applied. In addition, appropriate semi-implicit treatments are adopted to the convection terms, for both the velocity vector and orientation vector, as well as the the coupled elastic stress terms. In turn, all the semi-implicit terms could be represented as a linear operator of a vector potential, and its combination with the convex splitting discretization for the penalty function leads to a unique solvability analysis for the proposed numerical scheme. Furthermore, a careful estimate reveals an unconditional energy stability of the numerical system, composed of the kinematic energy and internal elastic energies. More importantly, an optimal rate convergence analysis and error estimate for the numerical scheme, which will be the first such result in the area.

Dr. Cheng Wang is a professor in Department of  Mathematics at the University of Massachusetts Dartmouth (UMassD). He obtained his Ph.D degree from Temple University in 2000, under the supervision of  Prof. Jian-Guo Liu. Prior to joining UMassD in 2008 as an assistant professor, he was a Zorn postdoc at Indiana University from 2000 to 2003, under the supervision of Roger Temam and Shouhong Wang, and he worked as an assistant professor at University of Tennessee at Knoxville from 2003 to 2008. Dr. Wang’s research interests include development of stable, accurate numerical algorithms for partial differential equations and numerical analysis. He has published more than 80 papers with more than 3600 citations. He also serves in the Editorial Board of “Numerical Mathematics: Theory, Methods and Applications”.