Tridiagonal matrix algorithm

Abstract: The alternating direction implicit (ADI) technique is used in the numerical solution scheme for each longitudinal step, in the well-known vectorial beam propagation method (VBPM). It is used to reduce the linear-system matrix into a set of tridiagonal ones, which can then be solved by Thomas algorithm. Solutions using standard (Crank-Nicolson) VBPM for step-index waveguides contain small oscillations, which can significantly reduce the accuracy of the solutions. We show that this "noise" can be removed by a properly tailored low-pass digital filter in the spatial frequency domain. Results become as accurate as those obtained by iterative techniques. Examples deal first with the scaler LP/sub 01/ mode and a Gaussian field in a single-mode fiber converging to the vector HE/sub 11/ mode. Second, with an optimized Ti:LiNbO/sub 3/ bent waveguide using MgO. Finally, with a rib waveguide where advantages related to the use of the digital filter become very evident.

Abstract: Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with orders and are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes.