Finite difference method

Abstract: In this paper we shall propose a simple scheme for calculating Green's functions for photons propagating in complex structured dielectrics or other photonic systems The method is based on an extension of the finite-difference time-domain (FDTD) method, originally proposed by Yee [IEEE Trans Antennas Propag 14, 302 (1966)], also known as the order-$N$ method [Chan, Yu, and Ho, Phys Rev 51, 16 635 (1995)] which has recently become a popular way of calculating photonic band structures We give a transparent derivation of the order-$N$ method which, in turn, enables us to give a simple yet rigorous derivation of the criterion for numerical stability as well as statements of charge and energy conservation which are exact even on the discrete lattice We implement this using a general, nonorthogonal coordinate system without incurring the computational overheads normally associated with nonorthogonal FDTD We present results for local densities of states calculated using this method for a number of systems First, we consider a simple one-dimensional dielectric multilayer, identifying the suppression in the state density caused by the photonic band gap and then observing the effect of introducing a defect layer into the periodic structure Second, we tackle a more realistic example by treating a defect in a crystal of dielectric spheres on a diamond lattice This could have application to the design of superefficient laser devices utilizing defects in photonic crystals as laser cavities

Abstract: A method is presented for computing deformation of very soft tissue. The method is motivated by the need for simple, automatic model creation for real-time simulation. The method is meshless in the sense that deformation is calculated at nodes that are not part of an element mesh. Node placement is almost arbitrary. Fully geometrically nonlinear Total Lagrangian formulation is used. Geometric integration is performed over a regular background grid that does not conform to the simulation geometry. Explicit time integration is used via the central difference method. As an example the simple but fully nonlinear Neo-Hookean material model is employed. The results are compared with a finite element simulation to verify the usefulness of the method. Copyright © 2010 John Wiley & Sons, Ltd.

Abstract: A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.

Abstract: THE MATHEMATICAL MODELS FOR FLUID FLOW SIMULATIONS AT VARIOUS LEVELS OF APPROXIMATION. The Basic Equations of Fluid Dynamics. The Dynamic Levels of Approximation. The Mathematical Nature of the Flow Equations and Their Boundary Conditions. BASIC DISCRETIZATION TECHNIQUES. The Finite Difference Method. The Finite Element Method. Finite Volume Method and Conservative Discretizations. THE ANALYSIS OF NUMERICAL SCHEMES. The Concepts of Consistency, Stability, and Convergence. The Von Neumann Method for Stability Analysis. The Method of the Equivalent Differential Equation for the Analysis of Stability. The Matrix Method for Stability Analysis. THE RESOLUTION OF DISCRETIZED EQUATIONS. Integration Methods for Systems of Ordinary Differential Equations. Iterative Methods for the Resolution of Algebraic Systems. Appendix. Index.