Finite difference method

Abstract: A family of P-stable high algebraic order exponentially-fitted methods for the numerical solution of the Schrodinger equation is developed in this paper Numerical illustration to the resonance problem of the radial Schrodinger equation indicates that the new proposed methods are generally more efficient than the previously developed exponentially-fitted methods of the same kind

Abstract: A three-dimensional numerical model based on the complete Navier-Stokes equations is developed and presented in this paper. The model can be used for the problem of propagation of fully nonlinear water waves. The Navier-Stokes equations are first transformed from an irregular calculation domain to a regular one using sigma coordinates. The projection method is used to separate advection and diffusion terms from the pressure terms in Navier-Stokes equations. MacCormack's explicit scheme is used for the advection and diffusion terms, and it has second-order accuracy in both space and time. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method. A staggered mesh and central spatial finite-difference scheme are used. The model is tested against the experimental data of Luth et al., and the comparison shows that higher harmonics can be modeled well. Comparison of the model solutions with the elliptic shoal case confirms that the present model works well for wave refraction and diffraction with strong wave focusing.

Abstract: An efficient and simple explicit finite difference beam propagation method (EFD-BPM) incorporating nonuniform mesh is described. The criteria for stability are developed, and it is shown that this algorithm is power conserving when the stability criteria are met. EFD-BPM is applied to the analysis of single and coupled semiconductor rib waveguides and its accuracy is confirmed by comparing the results with the reported results. Nonuniform mesh is found to improve the efficiency of the method significantly for the analysis of weakly guiding waveguide structures. Several coupled rib waveguide structures with curved input and output branching sections are analyzed using both three-dimensional EFD-BPM and two-dimensional finite difference BPM combined with effective index approximation. >