Numerical Techniques for Schroedinger Equations in the Semiclassical Regime
报告人:
Peter A. Markowich
教授
报告时间:
2010-04-22 15:30
报告地点:
理科楼1304报告厅
主办单位:
数学系
简介:
摘要：Linear (and nonlinear) Schroedinger equations in the semiclassical (small dispersion) regime pose a significant challenge to numerical analysis and scientific computing, mainly due to the fact that they propagate high frequency spatial and temporal oscillations. At first we prove using Wigner measure techniques that finite difference discretizations in general require a disproportionate amount of computational resources, since underlying numerical meshes need to be fine enough to resolve all oscillations of the solution accurately, even if only accurate observables are required. This can be mitigated by using a spectral (in space) discretization, combined with appropriate time splitting. Such discretizations are time-transverse invariant and allow for much coarser meshes than finite difference discretizations.In many physical applications highly oscillatory periodic potentials occur in Schroedinger equations, still aggravating the oscillatory solution structure. For such problems we present a numerical method based on the Bloch decomposition of the wave function.
