Numerical Solutions from Highly Oscillatory Integrals to Highly Oscillatory PDEs

  • Speaker(s)Qin Sheng (Baylor University)
  • DateFrom 2014-06-16 To 2014-06-16
  • VenueRoom 82J04, Jia Yi Bing Building, 82 Jing Chun Yuan, BICMR

Speaker: Qin Sheng (Baylor University)

Time: Mon, 06/16/2014 - 14:00

Place: Room 82J04, Jia Yi Bing Building, 82 Jing Chun Yuan, BICMR

Abstract: Highly oscillatory integrals have an unwarranted reputation being difficult to solve efficiently. However, such integrals have numerous important applications from signal analysis to laser beam propagations. On the other hand, similar concerns exist for solving linear or nonlinear wave equations when wave numbers involved are exceptionally large. In the case, merits of a conventional, and normally highly efficient, splitting method may diminish due to the fact that tiny discretization steps need to be employed to compensate high oscillations. Based on several strategies for solving oscillatory integrals and in geometric optics, we introduce an alternative way for solving highly oscillatory paraxial wave problems via a modified splitting strategy. In the process, an exponential transformation is first introduced to convert the underlying differential equation to coupled nonlinear equations. Then the equations are approximated by an oscillation-free semidiscretized system which is then treated by a Local-One-Dimensional (LOD) procedure for desired accuracy, efficiency and computability. The splitting method acquired is asymptotically stable and easy to use. Some computational experiments will be presented to illustrate our results and wave simulations.