In this thesis, we study the fast Huygens sweeping method for Schrödinger equations. We propose a simple backward time marching step to avoid the limitation on the timestep. We the apply the modication to simulate the multi-color optical self-focusing phenomena in nematic liquid crystals. The propagation of the nematicon is modeled by a parabolic wave equation coupled with a nonlinear elliptic partial differential equation governing the angle between the crystal and the direction of propagation. Numerically, the paraxial parabolic wave equation is solved by a fast Huygens sweeping method, while the nonlinear elliptic PDE is handled by the alternating direction explicit (ADE) method. The overall algorithm is shown to be numerically efficient for computing high frequency beam pro...[ Read more ]

In this thesis, we study the fast Huygens sweeping method for Schrödinger equations. We propose a simple backward time marching step to avoid the limitation on the timestep. We the apply the modication to simulate the multi-color optical self-focusing phenomena in nematic liquid crystals. The propagation of the nematicon is modeled by a parabolic wave equation coupled with a nonlinear elliptic partial differential equation governing the angle between the crystal and the direction of propagation. Numerically, the paraxial parabolic wave equation is solved by a fast Huygens sweeping method, while the nonlinear elliptic PDE is handled by the alternating direction explicit (ADE) method. The overall algorithm is shown to be numerically efficient for computing high frequency beam propagations. Finally, we further extend the method to the nonlocal nonlinear Schrödinger equation which model the interaction of particles in the Bose-Einstein condensations (BECs).