Topic
Finite difference method
About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: In this article, a general numerical method to solve two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically is presented.
Abstract: A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays.
A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology.
In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.
136 citations
TL;DR: In this paper, a program package was developed to evaluate electromagnetic fields inside arbitrary transmission-line connecting structures and to compute the scattering matrix, and detailed results were given and discussed regarding the fundamental behavior of embedding.
Abstract: The embedding of microwave devices is treated by applying the finite-difference method to three-dimensional shielded structures. A program package was developed to evaluate electromagnetic fields inside arbitrary transmission-line connecting structures and to compute the scattering matrix. The air bridge, the transition through a wall, and the bond wire are examined as interconnecting structures. Detailed results are given and discussed regarding the fundamental behavior of embedding.
136 citations
TL;DR: In this paper, a study of the errors in out-of-plane vorticity calculated using a local χ2 fitting of the measured velocity field and analytic differentiation has been carried out.
Abstract: A study of the errors in out-of-plane vorticity (ω
z
) calculated using a local χ2 fitting of the measured velocity field and analytic differentiation has been carried out. The primary factors of spatial velocity sampling separation and random velocity measurement error have been investigated. In principle the ω
z
error can be decomposed into a bias error contribution and a random error contribution. Theoretical expressions for the transmission of the random velocity error into the random vorticity error have been derived. The velocity and vorticity field of the Oseen vortex has been used as a typical vortex structure in this study. Data of different quality, ranging from exact velocity vectors of analytically defined flow fields (Oseen vortex flow) sampled at discrete locations to computer generated digital image frames analysed using cross-correlation DPIV, have been investigated in this study. This data has been used to provide support for the theoretical random error results, to isolate the different sources of error and to determine their effect on ω
z
measurements. A method for estimating in-situ the velocity random error is presented. This estimate coupled with the theoretically derived random error transmission results for the χ2 vorticity calculation method can be used a priori to estimate the magnitude of the random error in ω
z
. This random error is independent of a particular flow field. The velocity sampling separation is found to have a profound effect on the precise determination of ω
z
by introducing a bias error. This bias error results in an underestimation of the peak vorticity. Simple equations, which are based on a local model of the Oseen vortex around the peak vorticity region, allowing the prediction of the ω
z
bias error for the χ2 vorticity calculation method, are presented. An important conclusion of this study is that the random error transmission factor and the bias error cannot be minimised simultaneously. Both depend on the velocity sampling separation, but with opposing effects. The application of the random and bias vorticity error predictions are illustrated by application to experimental velocity data determined using cross-correlation DPIV (CCDPIV) analysis of digital images of a laminar vortex ring.
136 citations
TL;DR: Theory of finite difference methods for approximating the solutions of partial differential equations of parabolic type and unconditionally unstable difference equations and higher order correct difference equations are introduced, and a comparison of the calculation requirements is presented.
Abstract: Publisher Summary The chapter introduces the theoretical view of finite difference methods for approximating the solutions of partial differential equations of parabolic type. A few preliminary definitions and facts about difference analogues of derivatives are first presented. The symbols “u” and “w” will be used to denote the solution of a differential equation and the solution of a difference equation, respectively. Numerical treatment of parabolic differential equations is done by considering the boundary value problem for the heat equation in one space variable. The chapter begins by deriving the backward difference equation and the Crank-Nicolson difference equation. The local error in the time direction is decreased by deriving the Crank-Nicolson difference equation. Crank-Nicolson equations can be applied to problems for which slope conditions are specified at a boundary, but the disadvantage of the Crank-Nicolson method is that greater smoothness is required of the solution of the differential equation to insure convergence. The chapter deals with unconditionally unstable difference equations and higher order correct difference equations, and presents a comparison of the calculation requirements.
136 citations
TL;DR: A mathematical model for coupled heat and moisture transfer problem is presented in this article, which consists of four partial differential equations for mass balance, heat balance and drying rate of grain drying.
Abstract: Grain drying is a simultaneous heat and moisture transfer problem. The modelling of such a problem is of significance in understanding and controlling the drying process. In the present study, a mathematical model for coupled heat and moisture transfer problem is presented. The model consists of four partial differential equations for mass balance, heat balance, heat transfer and drying rate. A simple finite difference method is used to solve the equations. The method shows good flexibility in choosing time and space steps which enable the simulation of long term grain drying/cooling processes. A deep barley bed is used as an example of grain beds in the current simulation. The results are verified against experimental data taken from literature. The analysis of the effects of operating conditions on the temperature and moisture content within the bed is also carried out
136 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 53 |
| 2024 | 136 |
| 2023 | 170 |
| 2022 | 357 |
| 2021 | 733 |
| 2020 | 690 |