Report title: A structure preserving difference scheme with fast algorithms for high dimensional nonlinear space-fractional Schrödinger Equations

Reporter: Professor Liu Yang, School of Mathematical Sciences, Inner Mongolia University

Reporting time: 10:50-11:30 am, November 19, 2020

Report location: Tencent Conference ID: 892 255 965

Conference password: 1119

School contact: Lu Junliang lvjl@jlu.edu.cn

Report summary:

    In this talk, we aim at analyzing a novel structure preserving difference scheme for the high dimensional nonlinear space-fractional Schrödinger equation. The temporal direction is discretized by the modified Crank-Nicolson scheme, while the spacial variable is approximated by formulas from the shifted convolution quadrature. The energy and mass preserving properties are proved rigorously for the scheme based on the newly developed properties of the coefficients matrix. Further, the optimal convergence rate O (h2 +τ2) is derived in detail where h, τ denote the spacial and temporal mesh sizes, respectively. To save the computing cost, we consider the preconditioned fast BiCG (PF-BiCG) algorithm for the resulting complex systems. Numerical experiments are conducted to confirm our theoretical conclusions and the efficiency of the fast algorithm.

Brief introduction of the speaker:

   Liu Yang is a professor and doctoral supervisor in the School of Mathematical Sciences of Inner Mongolia University, and concurrently a director of the Computational Mathematics Branch of the Chinese Mathematical Society. From 2017.7 to 2018.7, visiting scholars from Queensland University of Technology in Australia and visiting Beijing Computing Science Research Center from time to time. Mainly engaged in the research of standard finite element method, mixed element method, finite difference method and discontinuous Galerkin method of differential equations and fractional differential equations. Currently in "Journal of Computational Physics", "International Journal of Heat and Mass Transfer", "Nonlinear Dynamics", "Numerical Methods for Partial Differential Equations", "Applied Mathematics Letters", "Applied Mathematics and Computation", "Computers & Mathematics" With Applications", "Numerical Algorithms" and other journals have published many academic papers, and published two monographs in National Defense Industry Press and Xidian University Press. At present, he is in charge of many projects including 3 National Natural Science Foundation of China in research or finalization. Selected into the first level of young innovative and entrepreneurial talents in the "Grassland Talents" project in the autonomous region.