Optical Waveguide Modelling Based On Scalar Finite Difference Scheme

A Numerical Method Based On Scalar Finite Difference Scheme Was Adopted And Programmed On MATLAB® Platform For Optical Waveguide Modeling Purpose. Comparisons With Other Established Methods In Terms Of Normalized Propagation Constant Were Included To Verify Its Applicability. The Comparison Results Obtained Were Proven To Have The Same Qualitative Behaviour. Furthermore, The Performances Were Evaluated In Terms Of Computation Complexity, Mesh Size, And Effect Of Acceleration Factor. Computation Complexity Can Be Reduced By Increasing The Mesh Size Which Will Then Produce More Error. The Problem Can Be Rectified By Introducing The Acceleration Factor, Successive Over Relaxation (SOR) Parameter. It Shows That SOR Range Between 1.3 And 1.7 Can Give Shorter Computation Time, While Producing Constant Value Of Simulation Results

A Conservative Finite Difference Scheme for Poisson-Nernst-Planck Equations

A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson-Nernst-Planck(PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, which is second-order accurate in both space and time. We use the physical parameters specifically suited toward the modelling of ion channels. We present a simple iterative scheme to solve the system of nonlinear equations resulting from discretizing the equations implicitly in time, which is demonstrated to converge in a few iterations. We place emphasis on ensuring numerical methods to have the same physical properties that the PNP equations themselves also possess, namely conservation of total ions and correct rates of energy dissipation. We describe in detail an approach to derive a finite-difference method that preserves the total concentration of ions exactly in time. Further, we illustrate that, using realistic values of the physical parameters, the conservation property is critical in obtaining correct numerical solutions over long time scales

An explicit finite difference scheme for the Camassa-Holm equation

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.Comment: 45 pages, 6 figure