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.
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Papers
Book•
1 Jan 1969
TL;DR: In this paper, the authors present an extension of Matrix Stability Analysis to the case of Parabolic Equations (see Section 2.1.1). But they do not discuss the use of the concept of a simple explicit method.
Abstract: Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.
275 citations
Esso1
TL;DR: In this article, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity for convection-diffusion equations and the primary purpose of this study is to give the user more than just a qualitative feel for the importance of truncation errors.
Abstract: Truncation error limits the use of numerical finite difference approximations to solve partial differential equations. In the solution of convection-diffusion equations such as occur in miscible displacement and thermal transport, truncation error results in an artificial dispersion term often denoted as numerical diffusion. The differential equations describing 2-phase fluid flow can also be rearranged into a convection-diffusion form. Miscible and immiscible differential equations have been shown to be completely analogous. In this form, it is easy to infer that numerical diffusion will result in an additional term resembling flow due to capillarity. Many users of numerical programs and probably all numerical analysts recognize that the magnitude of the numerical diffusivity for convection-diffusion equations can depend on both block size and time step. Most expressions developed in the literature have been used primarily to determine the order of the error rather than to quantify it. The primary purpose of this study is to give the user more than just a qualitative feel for the importance of truncation error. Insofar as possible, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity. (14 refs.)
275 citations
Book•
25 Jun 2004
TL;DR: In this article, the authors present a two-step Exact Difference Scheme and its applications, as well as two-stage Difference Schemes Generated by Taylor's Decomposition.
Abstract: 1 Linear Difference Equations.- 1.1 Difference Equations of the First Order.- 1.2 Difference Equations of the Second Order.- 1.3 Difference Equations with Constant Coefficients.- 2 Difference Schemes for First-Order Differential Equations.- 2.1 Single-Step Exact Difference Scheme and Its Applications.- 2.2 Taylor's Decomposition on Two Points and Its Applications.- 3 Difference Schemes for Second-Order Differential Equations.- 3.1 Two-Step Exact Difference Scheme and Its Applications.- 3.2 Taylor's Decomposition on Three Points and Its Applications.- 4 Partial Differential Equations of Parabolic Type.- 4.1 A Cauchy Problem. Well-posedness.- 4.2 Difference Schemes Generated by an Exact Difference Scheme.- 4.3 Single-Step Difference Schemes Generated by Taylor's Decomposition.- 5 Partial Differential Equations of Elliptic Type.- 5.1 A Boundary-Value Problem. Well-posedness.- 5.2 Difference Schemes Generated by an Exact Difference Scheme.- 5.3 Two-Step Difference Schemes Generated by Taylor's Decomposition.- 6 Partial Differential Equations of Hyperbolic Type.- 6.1 A Cauchy Problem.- 6.2 Difference Schemes Generated by an Exact Difference Scheme.- 6.3 Two-Step Difference Schemes Generated by Taylor's Decomposition.- 7 Uniform Difference Schemes for Perturbation Problems.- 7.1 A Cauchy Problem for Parabolic Equations.- 7.2 A Boundary-Value Problem for Elliptic Equations.- 7.3 A Cauchy Problem for Hyperbolic Equations.- 8 Appendix: Delay Parabolic Differential Equations.- 8.1 The Initial-Value Differential Problem.- 8.2 The Difference Schemes.- Comments on the Literature.
274 citations
TL;DR: In this article, a rigorous analysis of the numerical error associated with the use of stair-stepped (saw-toothed) approximation of a conducting boundary for finite-difference time-domain (FDTD) simulations is presented.
Abstract: A rigorous analysis of the numerical error associated with the use of stair-stepped (saw-toothed) approximation of a conducting boundary for finite-difference time-domain (FDTD) simulations is presented. First, a dispersion analysis in two dimensions is performed to obtain the numerical reflection coefficient for a plane wave scattered by a perfectly conducting wall, tilted with respect to the axes of the finite-difference grid, under both transverse electric and transverse magnetic polarizations. The characteristic equation for surface waves that can be supported by such saw-tooth conducting surfaces is derived. This equation leads to expressions that show the dependence of the propagation constant along the boundary and the attenuation constant perpendicular to it on cell size and wavelength. Numerical simulations that demonstrate the effects predicted by the dispersion analysis are presented. >
270 citations
TL;DR: In this paper, the stability of boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier-Stokes equations for laminar two-dimensional flows.
Abstract: The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.
269 citations
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