Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

The Particle-In-Cell Method for Hydrodynamic Calculations

Abstract: This report supersedes LAMS-1956 and LAMS-2052. A method is presented for solving hydrodynamic problems involving large distortions and compressions of the fluid in several space dimensions. The calculation proce- dure introduces finite difference approximations to the differential equations; the solution in practice is carried out by means of high-speed electronic computers. The characteristics of the method are discussed and these are illustrated by presenting results of representative calculations. (auth)

A Central Difference Numerical Scheme for Fractional Optimal Control Problems

Abstract: This paper presents a modified numerical scheme for a class of Frac- tional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub- domains, and a fractional derivative (FDs) at a time node point is approx- imated using a modified Grunwald-Letnikov approach. For the first order derivative, the proposed modified Grunwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the Fractional Optimal Control (FCO) equations, it leads to a set of alge- braic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system, and 2) as the sizes of the sub- domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.

Calculation of Stiffness and Damping Coefficients for Elastically Supported Gas Foil Bearings

Abstract: The stiffness and damping coefficients of an elastically supported gas foil bearing are calculated. A perfect gas is used as the lubricant, and its behavior is described by the Reynolds equation. The structural model consists only of an elastic foundation. The fluid equations and the structural equations are coupled. A perturbation method is used to obtain the linearized dynamic coefficient equations. A finite difference formulation has been developed to solve for the four stiffness and the four damping coefficients. The effect of the bearing compliance on the dynamic coefficients is discussed in this paper.

Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations

Abstract: The need often arises to analyze the dynamics of a system in terms of a discrete formulation. This can occur by using an a priori discrete model of the system or by discretizing a continuous model. For the latter case, the continuous model is represented by differential equations and the discrete forms come from the requirement to numerically integrate these equations. The concept of “dynamic consistency” plays an essential role in the construction of such discrete models which usually are expressed as finite difference equations. We define this concept and illustrate its application to the construction of nonstandard finite difference schemes.