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
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Papers
TL;DR: The combination of automatic variable order (via deferred corrections) and automatic (adaptive) mesh selection produces, as in the case of initial value problem solvers, a versatile, robust, and efficient algorithm.
Abstract: A variable order variable step finite difference algorithm for approximately solving m-dimensional systems of the form y'' = f(t,y), t $\in$ [a,b] subject to the nonlinear boundary conditions g(y(a),y(b)) = 0 is presented. A program, PASVAR, implementing these ideas has been written and the results on several test runs are presented together with comparisons with other methods. The main features of the new procedure are: a) Its ability to produce very precise global error estimates, which in turn allow a very fine control between desired tolerance and actual output precision. b) Non-uniform meshes allow an economical and accurate treatment of boundary layers and other sharp changes in the solutions. c) The combination of automatic variable order (via deferred corrections) and automatic (adaptive) mesh selection produces, as in the case of initial value problem solvers, a versatile, robust, and efficient algorithm.
213 citations
TL;DR: In this paper, a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered, where the integral term is treated by means of a convolution quadrature suggested by Lubich.
Abstract: A difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered. The integral term is treated by means of a convolution quadrature suggested by Lubich. Some results from Lubich’s discretized fractional calculus play a crucial role in proving consistency. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. A stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper.
212 citations
TL;DR: A general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes for weak solutions containing nonclassical regularization-sensitive shock waves.
Abstract: We consider weak solutions of (hyperbolic or hyperbolic-elliptic) systems of conservation laws in one-space dimension and their approximation by finite difference schemes in conservative form. The systems under consideration are endowed with an entropy-entropy flux pair. We introduce a general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes. In general, these schemes are fully nonlinear implicit, but in some important cases can be explicit or linear implicit. Furthermore, semidiscrete entropy conservative schemes of arbitrary order are presented. The entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization-sensitive shock waves. Finally, specific examples are investigated and tested numerically. Our approach extends the results and techniques by Tadmor [in Numerical Methods for Compressible Flows---Finite Difference, Element and Volume Techniques, ASME, New York, 1986, pp. 149--158], LeFloch and Rohde [SIAM J. Numer. Anal., 37 (2000), pp. 2023--2060].
212 citations
TL;DR: In this paper, a new mathematical formulation for the systematic development of perfectly matched layers from Maxwell's equations in properly constructed anisotropic media is presented, which can be implemented in the time domain without any splitting of the fields.
Abstract: A new mathematical formulation is presented for the systematic development of perfectly matched layers from Maxwell's equations in properly constructed anisotropic media. The proposed formulation has an important advantage over the original Berenger's perfectly matched layer in that it can be implemented in the time domain without any splitting of the fields. The details of the numerical implementation of the proposed perfectly matched absorbers in the context of the finite-difference time-domain approximation of Maxwell's equations are given. Results from three-dimension (3-D) simulations are used to illustrate the effectiveness of the media constructed using the proposed approach as absorbers for numerical grid truncation.
211 citations
TL;DR: In this article, the authors used a three-dimensional implicit finite difference method with rectangular coordinate system to simulate the ground temperature and the borehole temperature profile for geothermal ground heat exchanger.
208 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 53 |
| 2024 | 136 |
| 2023 | 170 |
| 2022 | 357 |
| 2021 | 733 |
| 2020 | 690 |