Showing papers on "Finite difference published in 1981"
TL;DR: In this paper, highly absorbing boundary conditions for two-dimensional time-domain electromagnetic field equations are presented for both two-and three-dimensional configurations and numerical results are given that clearly exhibit the accuracy and limits of applicability of these boundary conditions.
Abstract: When time-domain electromagnetic-field equations are solved using finite-difference techniques in unbounded space, there must be a method limiting the domain in which the field is computed. This is achieved by truncating the mesh and using absorbing boundary conditions at its artificial boundaries to simulate the unbounded surroundings. This paper presents highly absorbing boundary conditions for electromagnetic-field equations that can be used for both two-and three-dimensional configurations. Numerical results are given that clearly exhibit the accuracy and limits of applicability of highly absorbing boundary conditions. A simplified, but equally accurate, absorbing condition is derived for two- dimensional time-domain electromagnetic-field problems.
2,553 citations
TL;DR: In this paper, the relevance of group velocity to the behavior of finite difference models of time-dependent partial differential equations is surveyed and illustrated, and applications involve the propagation of wave packets in one and two dimensions, numerical dispersion, the behaviour of parasitic waves, and the stability analysis of initial boundary value problems.
Abstract: The relevance of group velocity to the behavior of finite difference models of time-dependent partial differential equations is surveyed and illustrated. Applications involve the propagation of wave packets in one and two dimensions, numerical dispersion, the behavior of parasitic waves, and the stability analysis of initial boundary-value problems.
477 citations
TL;DR: In this article, one-sided or up-wind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws are analyzed and a second order scheme is designed for which they prove both nonlinear stability and that the entropy condition is satisfied for limit solutions.
Abstract: We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.
348 citations
TL;DR: In this article, a finite-difference method to approximate a Schrodinger equation with a power non-linearity is described, which is used to model the propagation of a laser beam in a plasma.
282 citations
Book•
01 Jan 1981
TL;DR: In this article, the authors present a review of the Finite Element Equation (FE) and its application to the 1-D and 2-D problems, including the P2-Triangle and the Q2-Quadrangle.
Abstract: Notations.- 1. Elliptic Equations of Order 2: Some Standard Finite Element Methods.- 1.1. A 1-Dimensional Model Problem: The Basic Notions.- 1.2. A 2-Dimensional Problem.- 1.3. The Finite Element Equations.- 1.4. Standard Examples of Finite Element Methods.- 1.4.1. Example 1: The P1-Triangle (Courant's Triangle).- 1.4.2. Example 2: The P2-Triangle.- 1.4.3. Example 3: The Q1-Quadrangle.- 1.4.4. Example 4: The Q2-Quadrangle.- 1.4.5. A Variational Crime: The P1 Nonconforming Element.- 1.5. Mixed Formulation and Mixed Finite Element Methods for Elliptic Equations.- 1.5.1. The One Dimensional Problem.- 1.5.2. A Two Dimensional Problem.- 2. Upwind Finite Element Schemes.- 2.1. Upwind Finite Differences.- 2.2. Modified Weighted Residual (MWR).- 2.3. Reduced Integration of the Advection Term.- 2.4. Computation of Directional Derivatives at the Nodes.- 2.5. Discontinuous Finite Elements and Mixed Interpolation.- 2.6. The Method of Characteristics in Finite Elements.- 2.7. Peturbation of the Advective Term: Bredif (1980).- 2.8. Some Numerical Tests and Further Comments.- 2.8.1. One Dimensional Stationary Advection Equation (56).- 2.8.2. Two Dimensional Stationary Advection Equation.- 2.8.3. Time Dependent Advection.- 3. Numerical Solution of Stokes Equations.- 3.1. Introduction.- 3.2. Velocity-Pressure Formulations: Discontinuous Approximations of the Pressure.- 3.2.1. uh: P1 Nonconforming Triangle ( 1-4-5) ph: Piecewise Constant.- 3.2.2. uh: P2 Triangle ph: P0 (Piecewise Constant).- 3.2.3. uh: "P2+bubble" Triangle (or Modified P2) ph: Discontinuous P1.- 3.2.4. uh: Q2 Quadrangle ph: Q1 Discontinuous.- 3.2.5. Numerical Solution by Penalty Methods.- 3.2.6. Numerical Results and Further Comments.- 3.3. Velocity-Pressure Formulations: Continuous Approximation of the Pressure and Velocity.- 3.3.1. Introduction.- 3.3.2. Examples and Error Estimates.- 3.3.3. Decomposition of the Stokes Problem.- 3.4. Vorticity-Pressure-Velocity Formulations: Discontinuous Approximations of Pressure and Velocity.- 3.5. Vorticity Stream-Function Formulation: Decompositions of the Biharmonic Problem.- 4. Navier-Stokes Equations: Accuracy Assessments and Numerical Results.- 4.1. Remarks on the Formulation.- 4.2. A review of the Different Methods.- 4.2.1 Velocity-Pressure Formulations: Discontinuous Approximations of the Pressure.- 4.2.2. Velocity-Pressure Formulations: Continuous Approximations of the Pressure.- 4.2.3. Vorticity-Pressure-Velocity Formulations: Discontinuous Approximations of Pressure and Velocity.- 4.2.4. Vorticity Stream-Function Formulation.- 4.3. Some Numerical Tests.- 4.3.1. The Square Wall Driven Cavity Flow.- 4.3.2. An Engineering Problem: Unsteady 2-D Flow Around and In an Air-Intake.- 5. Computational Problems and Bookkeeping.- 5.1. Mesh Generation.- 5.2. Solution of the Nonlinear Problems.- 5.2.1. Successive Approximations (or Linearization) with Under Relaxation.- 5.2.2. Newton-Raphson Algorithm.- 5.2.3. Conjugate Gradient Method (with Scaling) for Nonlinear Problems.- 5.2.4. A Splitting Technique for the Transient Problem.- 5.3. Iterative and Direct Solvers of Linear Equations.- 5.3.1. Successive Over Relaxation.- 5.3.2. Cholesky Factorizations.- 5.3.3. Out of Core Factorizations.- 5.3.4. Preconditioned Conjugate Gradient.- Appendix 2. Numerical Illustration.- Three Dimensional Case.- References.
217 citations
TL;DR: In this article, the authors compared two-dimensional finite element analysis with one-dimensional stochastic solutions for total and differential settlement, and found that the differences were mainly due to the randomness in the stress field which cannot be included in onedimensional models, and mechanistic correlations by common dependence on the realizations of particular random variables.
Abstract: Stochastic finite element analysis is used to predict uncertainties in total and differential settlement under a large flexible footing. The results are compared with one-dimensional stochastic solutions already in the literature. Differences between the one- and two-dimensional analyses, particularly for differential settlement, are distinct. These differences seem primarily attributable to randomness in the stress field which cannot be included in one-dimensional models, and to mechanistic correlations by common dependence on the realizations of particular random variables. In principle, second-moment techniques can be extended to a broad range of analyses now performed using finite element and finite difference techniques.
179 citations
TL;DR: In this paper, a new explicit, time splitting algorithm for finite difference modeling of the Navier-Stokes equations of fluid mechanics is presented. But it is not shown that the split operators achieve their maximum allowable time step, i.e., the corresponding Courant number.
151 citations
TL;DR: It is shown how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.
Abstract: The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.
123 citations
TL;DR: The finite-difference method is a direct, versatile, and reasonably efficient means of solving the two-dimensional cochlear model.
Abstract: A current, linear, two-dimensional mathematical model of the mechanics of the cochlea is solved numerically by using a finite difference approximation of the model equations. The finite-difference method is used to discretize Laplace's equation over a rectangular region with specified boundary conditions. The resulting matrix equation for fluid pressure is solved by using a Gaussian block-elimination technique. Numerical solutions are obtained for fluid pressure and basilar membrane displacement as a function of distance from the stapes. The finite difference method is a direct, versatile, and reasonably efficient means of solving the two-dimensional cochlear model.
118 citations
TL;DR: In this article, Petrov-Galerkin methods based on piecewise linear interpolants for the Korteweg-de Vries and related equations are studied and both accuracy and stability are analyzed for the linearised case.
115 citations
TL;DR: In this paper, a two-step expansion technique has been developed for finite-difference codes that can increase the spectral resolution of such codes over selected subvolumes of the original problem space by predicting the responses arising from the coupling of an incident exterior field across an aperture in a hollow cylinder to an interior wire.
Abstract: A two-step expansion technique has been developed for finite-difference codes that can increase the spectral resolution of such codes over selected subvolumes of the original problem space. The utility of this technique is demonstrated by predicting the responses arising from the coupling of an incident exterior field across an aperture in a hollow cylinder to an interior wire. Expansions on the order of fourfold or more are possible. The cost of such increased resolution is a second computer run, doubling the cost as opposed to a cost increase of a factor of 64 if the resolution equivalent to a fourfold expansion were sought from a single run. Possible applications extend to many diverse areas, but most importantly, interior coupling problems may be treated using this technique. For example, this technique may provide a useful method for estimating the responses of cables interior to an aircraft.
01 Jan 1981
TL;DR: This chapter discusses Computational Techniques for Solution of Convective Equations, which focuses on the application of Chebychev Iteration to Non-Self-Adjoint Equations.
Abstract: 1 Introduction.- 2 Computational Techniques for Solution of Convective Equations.- 2.1 Importance of Convective Equations.- 2.2 Requirements for Convective Equation Algorithms.- 2.3 Quasiparticle Methods.- 2.4 Characteristic Methods.- 2.5 Finite-Difference Methods.- 2.6 Finite-Element Methods.- 2.7 Spectral Methods.- 3 Flux-Corrected Transport.- 3.1 Improvements in Eulerian Finite-Difference Algorithms.- 3.2 ETBFCT: A Fully Vectorized FCT Module.- 3.3 Multidimensional FCT.- 4 Efficient Time Integration Schemes for Atmosphere and Ocean Models.- 4.1 Introduction.- 4.2 Time Integration Schemes for Barotropic Models.- 4.3 Time Integration Schemes for Baroclinic Models.- 4.4 Extension to Ocean Models.- 5 A One-Dimensional Lagrangian Code for Nearly Incompressible Flow.- 5.1 Difficulties Encountered in Lagrangian Methods.- 5.2 Adaptive Gridding in a Lagrangian Calculation.- 5.3 The Algorithm and Structure of ADINC.- 5.4 Examples.- 6 Two-Dimensional Lagrangian Fluid Dynamics Using Triangular Grids.- 6.1 Grid Distortion in Two Dimensions.- 6.2 Use of Reconnection to Eliminate Grid Distortion.- 6.3 Numerical Algorithms.- 6.4 Examples.- 7 Solution of Elliptic Equations.- 7.1 Survey of Standard Techniques.- 7.2 A New Direct Solver: The Stabilized Error Vector Propagation Technique (SEVP).- 7.3 Application of Chebychev Iteration to Non-Self-Adjoint Equations.- 8 Vectorization of Fluid Codes.- 8.1 Speed in Hardware.- 8.2 Speed in Fortran.- 8.3 Problems with Causality.- 8.4 Examples.- 8.5 Summary of Parallelism Principles.- Appendix A.- Appendix B.- Appendix C.- Appendix D.- Appendix E.- References.
TL;DR: In this paper, an implicit finite-difference (IFD) scheme is introduced for underwater acoustic wave propagation problems and applied to the parabolic wave equation model for the acoustic problem with a free surface and arbitrary bottom and bottom boundary conditions.
Abstract: In this article an implicit finite‐difference (IFD) scheme is introduced for solving underwater acoustic wave propagation problems. This scheme is applied to the parabolic wave equation model for the acoustic problem with a free surface and arbitrary bottom and bottom boundary conditions. The mathematical theory of the implicit finite‐difference method as applied to the parabolic equation is developed, and a computer model to implement the IFD is introduced. Representative problems are presented to demonstrate the capability of the IFD model. Advantages and limitations of the IFD are discussed.
TL;DR: In this article, it was shown that failure to take these singularities into account leads to large errors in the finite-difference solution of the time-domain electromagnetic field equations.
Abstract: When the electromagnetic-field equations are solved in a region with a corner, singularities in the field or in its spatial derivatives will be present at these corners. These singularities cause the load truncation error in a finite-difference approximation of the field equations to be unbounded. In this paper it is shown that failing to take these singularities into account leads to large errors in the finite-difference solution of the time-domain electromagnetic-field equations. A simple method is described to account for these singularities while retaining the simplicity of the finite-difference formulation. Numerical results are given that demonstrate the accuracy obtained when our technique is used.
TL;DR: In this article, simple upwind finite difference and element methods approximating nonlinear partial differential equations have been developed to approximate general systems of nonlinear hyperbolic conservation laws, nonlinear singular perturbation problems, and particular physical problems involving the equations of compressible fluid dynamics.
Abstract: We have recently, with Bjorn Engquist and various students, developed simple upwind finite difference and element methods approximating nonlinear partial differential equations. These methods are used to approximate (l) general systems of nonlinear hyperbolic conservation laws, (2) nonlinear singular perturbation problems, and (3) particular physical problems involving the equations of compressible fluid dynamics.
TL;DR: In this article, a theoretical model is presented to predict the moisture flow in an unsaturated soil as the result of hydraulic and temperature gradients, and a partial differential heat flow equation (for above-freezing conditions) and two partial differential transient flow equations (one for the water phase and the other for the air phase) are derived and solved using a finite difference technique.
Abstract: A theoretical model is presented to predict the moisture flow in an unsaturated soil as the result of hydraulic and temperature gradients. A partial differential heat flow equation (for above-freezing conditions) and the two partial differential transient flow equations (one for the water phase and the other for the air phase), are derived in this paper and solved using a finite difference technique. Darcy's law is used to describe the flow in the water phase, while Pick's law is used for the air phase. The constitutive equations proposed by Fredlund and Morgenstern are used to define the volume change of an unsaturated soil. The simultaneous solution of the partial differential equations gives the temperature, the pore water pressure, and the pore air pressure distribution with space and time in an unsaturated soil. The pressure changes can, in turn, be used to compute the quantity of moisture flow.
TL;DR: Stone's unconditionally stable, strongly implicit numerical method is extended to the 2 x 2 coupled vorticity-stream function form of the Navier-Stokes equations as mentioned in this paper, which allows for complete coupling of the boundary conditions.
TL;DR: Two one-parameter families of fourth order HODIE discretizations of the Helmholtz equation are derived and a discretization optimal with respect to a certain norm of the truncation error is identified.
Abstract: In this paper we construct and analyze high order finite difference discretizations of a class of elliptic partial differential equations. In particular, two one-parameter families of fourth order HODIE discretizations of the Helmholtz equation are derived and a discretization optimal with respect to a certain norm of the truncation error is identified. The use of compact nine-point formulas of positive type admits both fast direct methods and standard iterative methods for the solution of the resulting systems of linear equations. Extensions yielding sixth order accuracy for the Helmholtz equation and fourth order accuracy for a more general operator are given. Finally, numerical results demonstrating the effectiveness of the discretizations for a wide range of problems are presented.
TL;DR: In this article, the Galerkin finite element method is utilized to obtain quite detailed results for flow through a channel containing a step at Reynolds numbers of 0 and 200, however, this technique is prone to generating wiggles or oscillations when streamwise gradients become too large to be resolved by the mesh.
TL;DR: In this paper, the optimum design of a structure subject to temperature constraints is considered when mathematical optimization techniques are used, derivatives of the temperature constraints with respect to the design variables are usually required In the case of large aerospace structures, such as the Space Shuttle, the computation of these derivatives can become prohibitively expensive and a finite difference approach has been considered in studies conducted to improve the efficiency of the calculation of the derivatives.
Abstract: The optimum design of a structure subject to temperature constraints is considered When mathematical optimization techniques are used, derivatives of the temperature constraints with respect to the design variables are usually required In the case of large aerospace structures, such as the Space Shuttle, the computation of these derivatives can become prohibitively expensive Analytical methods and a finite difference approach have been considered in studies conducted to improve the efficiency of the calculation of the derivatives The present investigation explores two possibilities for enhancing the effectiveness of the finite difference approach One procedure involves the simultaneous solution of temperatures and derivatives The second procedure makes use of the optimum selection of the magnitude of the perturbations of the design variables to achieve maximum accuracy
TL;DR: A method is described which, through the introduction of a different set of network variables, significantly reduces the size of the original system, avoids the need to compute pressures, and produces velocities that are exactly discrete divergence free.
TL;DR: In this article, the authors proposed a nonlinear three-dimensional initial value computer code RSF for the evolution of resistive tearing modes in MHD activity and disruptions in tokamaks.
01 Jan 1981
TL;DR: FISHPAK is a package of FORTRAN subroutines that has been developed at the National Center for Atmospheric Research that provides a basic capability to automatically produce finite difference approximations to Helmholtz's equation defined on a rectangle in a particular co-ordinate system.
Abstract: Publisher Summary
This chapter discusses the efficient FORTRAN subprograms for the solution of elliptic partial differential equations. FISHPAK is a package of FORTRAN subroutines that has been developed at the National Center for Atmospheric Research. The package provides a basic capability to automatically produce finite difference approximations to Helmholtz's equation defined on a rectangle in a particular co-ordinate system. Also, there are routines for the more general separable elliptic equation and three-dimensional Helmholtz equation in Cartesian coordinates. The package is comprised of a set of drivers, a set of solvers, and an extensive subpackage for computing Fourier transforms. Drivers are available on two different types of finite difference grids, namely, centered, and staggered. In a centered grid, the physical boundaries coincide with the grid lines, whereas in a staggered grid, the physical boundaries are located at one-half grid spacing away from the grid lines. All drivers use a uniform or equally spaced grids in the given coordinate system.
TL;DR: In this paper, the errors due to developmental finite difference approximations in the two-microphone acoustic intensity measurement technique are considered and a lower limiting frequency for intensity measurements is determined to prevent these low frequency errors.
TL;DR: In this article, a reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates.
TL;DR: In this paper, the cylindrical polar coordinate system of nodes developed for the wedge tool has been found to be superior in mathematical accuracy and computer implementation to the cartesian co-ordinate systems commonly used in finite difference temperature methods.
TL;DR: In this article, the authors extend the discussion in [9] concerning the 'numerically irrelevant' solutions NIS' of finite difference approximations to certain nonlinear boundary value problems in the context of continuation methods or more generally and more appropriately, in the case of global topological perturbations of nonlinear eigen value problems.
Abstract: The aim of this paper is to continue and extend the discussion in [9] concerning the 'numerically irrelevant' Solutions NIS' of finite difference approximations to certain nonlinear boundary value problems in the context of continuation methods or more generally and more appropriately, in the context of global topological perturbations of nonlinear eigen value problems. This latter technique äs a numerical device for the global numerical study of nonlinear eigenvalue and bifurcation problems has been set up, discussed and applied to various nonlinear problems in [9] and [11]. Our discussion here will make an essential use of these techniques and ideas and moreover, provide a mathematical foundation for the numerical procedures suggested in [9] which were designed to avoid NIS. Our present attempt and approach has been motivated by a recent numerical study of Bohl [4] and a recent paper of Ambrosetti and Hess [3]. The former deals with the existence and characterization of NIS of finite difference approximations to a boundary value problem of the type
TL;DR: In this article, a moving, deforming coordinate system (MDCS) is applied to the standard transport equation test case, and great efficiency and accuracy are obtained in early time, using no more nodes and time steps overall than during early time runs.
Abstract: Transport problems with sharp transitions or high convection often stress even the most advanced computational methods beyond feasibility limits. One promising line of attack involves transforming problems to a moving, deforming coordinate system (MDCS). Using MDCS as recently developed, one may concentrate computational attention when and where it is most appropriate, also possibly greatly reducing the effective magnitude of terms which cause trouble. In principle, MDCS may be used with finite difference, finite element, or other numerical methods, according to preference. In the particular finite element system advanced here, mesh movement may be variable in time and space, so that physically fixed boundaries may remain fixed in the transformed coordinate system. A simple finite difference system is used in time. When the method is applied to the standard transport equation test case, great efficiency and accuracy are obtained in early time. Even very steep fronts may be represented well, with no oscillations, using space and time step sizes well beyond conventional (Peclet, Courant number) constraints. More important problems are solved, involving transport over much longer times and larger spaces, when diffusion has some chance to operate and hence a parabolic model is more warranted. The same accuracy and stability are achieved as during early time, using no more nodes and time steps overall than during early time runs. This constitutes orders of magnitude savings in effort over what the application of conventional constraints would require. Relative efficiency from application of MDCS to other problems would depend on the particulars of each case.
TL;DR: In this paper, the finite difference boundary value method with a complex rotated coordinate is used to obtain the resonances of the one-dimensional Schrodinger equation, which yields resonances with widths exceeding the real part of the energy.
TL;DR: In this article, a finite difference scheme for modeling the convection/diffusion equation in strongly convective flow regimes including circumstances in which significant source terms are present is proposed, and an assessment of its accuracy is made by means of a Taylor expansion analysis and a study of its performance in two model problems.
Abstract: This paper considers a finite difference scheme for modelling the convection/diffusion equation in strongly convective flow regimes including circumstances in which significant source terms are present.
The main objective is to provide an alternative approach to central and/or upwind difference methods which for various reasons are unsatisfactory. To illustrate the main features of the scheme, an assessment of its accuracy is made by means of a Taylor expansion analysis and a study of its performance in two model problems. As a demonstration of its generality for use in large-scale practical problems, some numerical results are presented for the prediction of the temperature distribution in a flow through a partially blocked heated rod bundle.
The main conclusions are that in almost all practical circumstances results obtained using the scheme are not susceptible to false diffusion or spatial oscillations, which are, respectively, the inherent weaknesses in many upwind and central difference scheme formulations, and in general its use results in improved overall accuracy.