Showing papers on "Finite difference coefficient published in 1995"

Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation

Abstract: In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some ...

A finite difference approximation of the kinematic wave model of traffic flow

Abstract: This article shows that if the kinematic wave model of freeway traffic flow in its general form is approximated by a particular type of finite difference equation, the finite difference results converge to the kinematic wave solution despite the existence of shocks in the latter. This result, which applies to initial and boundary condition problems with and without discontinuous data, is shown not to hold for other commonly used finite difference schemes. In the proposed approximation, the flow between two neighboring lattice points is the minimum of the two values returned by: 1. (a) a “sending” function evaluated at the density prevailing at the upstream lattice point and 2. (b) a “receiving” function evaluated at the downstream lattice point. The sending and receiving functions correspond to the increasing and decreasing branches of the freeway's flow-density curve. The article presents an asymptotic formula for the errors introduced by the proposed finite difference approximation and describes quantitatively the finite difference's shock-capturing behavior. Errors are shown to be approximately proportional to the mesh spacing with a coefficient of proportionality that depends on the wave speed, on its rate of change with density, and on the slope and curvature of the initial density profile. The asymptotic errors are smaller than those of Lax's first-order, centered difference method which is also convergent. More importantly though, the proposed procedure never yields negative flows, and this makes it attractive in practical engineering applications when the mesh cannot be made arbitrarily small.

A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation

Abstract: The Cahn-Hilliard equation is a nonlinear evolutionary equation that is of fourth order in space. In this paper a linearized finite difference scheme is derived by the method of reduction of order. It is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. The coefficient matrix of the difference system is symmetric and positive definite, so many well-known iterative methods (e.g. Gauss-Seidel, SOR) can be used to solve the system.

Numerical simulation of elastic wave propagation using a finite volume method

Abstract: Like the finite difference method, the finite volume method gives an approximate value for the derivative of a field at a given point using the values of the field at a few locations neighboring the point. The method uses the divergence theorem, considers a “finite volume” around the point and discretizes the surface bounding the volume. When the finite volumes considered are regular polyhedra, one obtains the expressions corresponding to standard centered finite differences, but the finite volume method is more general than the finite difference method because it may deal directly with irregular grids. It is possible to give a finite volume formulation of the elastodynamic problem, using dual volumes, that correspond, in the regular case, to the staggered grids used in the finite difference method. The scheme thus obtained is more general than the one obtained using finite differences, as the “grids” may be totally unstructured, but at the cost of having, in the general case, only a first-order accuracy. Although the scheme is not consistent, numerical tests suggest that it is stable and convergent. This implementation of a finite volume method does not provide a way for a more general treatment of the boundaries than the conventional finite difference method.

Finite difference reaction diffusion equations with nonlinear boundary conditions

Abstract: This article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time-dependent and steady-state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existence-comparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time-dependent solution in relation to the steady-state solutions. Application is given to a heat-conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady-state solutions, and determines the dynamic behavior of the time-dependent solution. Numerical results for the heat-conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. © 1995 John Wiley & Sons, Inc.

Generalized Finite Difference Methods for Optimal Estimation of Derivatives in Real-Time Control Problems:

Abstract: The real-time estimation of (particularly first and second) derivatives of motion from position data has many applications in automatic control. Errors in such estimates at high sampling rates aris...

On Taylor weak statement finite element methods for computational fluid dynamics

Abstract: SUMMARY ATaylor series augmentation of a weak statement (a ‘Taylor weak statement’ or ‘Taylor-Galerkin’ method) is used to systematically reduce the dispersion error in a finite element approximation of the one-dimensional transient advection equation. A frequency analysis is applied to determine the phase velocity of semi-implicit linear, quadratic and cubic basis one-dimensional finite element methods and of several comparative finite difference/ finite volume algorithms. The finite element methods analysed include both GaIerkin and Taylor weak statements. The frequency analysis is used to obtain an improved linear basis Taylor weak statement finite element algorithm. Solutions are reported for verification problems in one and two dimensions and are compared with finite volume solutions. The improved finite element algorithms have sufficient phase accuracy to achieve highly accurate linear transient solutions with little or no artificial diffusion. Application of the Galerkm finite element method (FEM) to parabolic and hyperbolic differential equations has presented difficulties with the control of dispersion error. Dispersion error results fiom shorter-wavelength solution components travelling at the wrong speed, usually too slowly. Waves travelling at the wrong speed eventually appear in the wrong place as extraneous short waves and can lead to instability in a non-linear problem statement. The ‘upwind’ finite volume method has been extensively applied either to reduce dispersion error or to artificially &&se the resulting short waves. The interpolation is biased for greater contribution fiom the direction of the velocity. It originated with the donor cell method of Courant et af. in 1952 and was applied to the FEM in the 1970s as the Petrov-Galerkin methods of Christie et ~1.~9 and Heinrich et ~1.~3~ Early Petrov-Galerkin methods suffered from excess diffusion of the solution and later work has been oriented towards reducing the excess diffusion, including the SUPG method of Brooks and Hughes6 in 1980 and the methods of Dick7 in 1983, Westerink and Sheas in 1989, Bouloutas and Celia9 in 199 1 and Konda et al. lo in 1992. Similar work in the finite volume method is typified by the QUICK methods of Leonard” in 1979 and Leonard and Mokhtari12 in 1992. Many upwind methods still cause excess diffusion and some also cause an undesirable increase in the width of the matrix stencil, thus increasing the computational effort. An alternative approach which avoids these problems originated for the finite volume method with Lax and WendroP3 in 1960. The

A fourth-order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations

Abstract: A finite difference scheme for the two-dimensional, second-order, nonlinear elliptic equation is developed. The difference scheme is derived using the local solution of the differential equation. A 13-point stencil on a uniform mesh of size h is used to derive the finite difference scheme, which has a truncation error of order h4. Well-known iterative methods can be employed to solve the resulting system of equations. Numerical results are presented to demonstrate the fourth-order convergence of the scheme. © 1995 John Wiley & Sons, Inc.

Learning in recurrent finite difference networks

Abstract: A recurrent learning algorithm based on a finite difference discretization of continuous equations for neural networks is derived. This algorithm has the simplicity of discrete algorithms while retaining some essential characteristics of the continuous equations. In discrete networks learning smooth oscillations is difficult if the period of oscillation is too large. The network either grossly distorts the waveforms or is unable to learn at all. We show how the finite difference formulation can explain and overcome this problem. Formulas for learning time constants and time delays in this framework are also presented.

Finite Difference Modeling of Anisotropic Flows

Abstract: A modification to the finite difference equations is proposed in modeling multidimensional flows in an anisotropic material. The method is compared to the control volume version of the Taylor expansion and the finite element formulation derived from the Galerkin weak statement. For the same number of nodes, the proposed finite difference formulation approaches the accuracy of the finite element method. For the two-dimensional case, the effect on accuracy and solution stability is approximately the same as quadrupling the number of nodes for the Taylor expansion with only a proportionately small increase in the number of computations. Excellent comparisons are made with a new limiting case exact solution modeling anisotropic heat conduction and a transient, anisotropic conduction experiment from the literature.

Maximum-angle condition and triangular finite elements of Hermite type

Abstract: Various triangular finite C°-elements of Hermite type satisfying the maximum-angle condition are presented and corresponding finite element interpolation theorems are proved. The paper contains also a proof that very general hypotheses due to Jamet are not necessary for such finite elements.

The remainder-effect analysis of finite difference schemes and the applications

Abstract: In the present paper, two contents are enclosed. First. the Fourier analysis approac.h of the dispersion relation and group velocio, effect of .finite difference schemes is discussed, the defects of the approach is pointed out and the correction is made; Second, a new systematic analysis method-remaider-effect analysis (abbr. REAM) is proposed by means of the modified partial differential equations (abbr. MPDE) of finite difference schemes. The analysis is based on the synthetical stud), of the rational dispersion-and dissipation relations of finite difference schemes. And the method clearly possesses constructivity.

Electromagnetic Scattering from Impedance Elliptic Cylinders Using Finite Difference Method (Oblique Incidence)

Abstract: The electromagnetic scattering from elliptic impedance cylinders illuminated by an obliquely incident plane wave is presented. The finite difference technique is used to solve this problem. Impedance Boundary Condition (IBC) is enforced on the cylinder surface and an Absorbing Boundary Condition (ABC) is applied on the outer boundary. Between the cylinder surface boundary and the outer boundary, the wave equation is solved with the finite difference technique to obtain the scattered field. Two methods are used to solve this problem and comparison between both methods is used to verify the solution. One method is based on transforming the elliptic cylinder into circular cylinder in polar coordinates. The other method is the direct implementation of the finite difference method in the elliptic coordinates. Consequently, proper transformation of IBC, ABC and wave equation is used. The solution obtained from both methods are found to be in good agreement with each other.

On the behavior of attractors under finite difference approximation

Abstract: We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C 1 perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.

Computation of discrete logarithms in an arbitrary finite field

Abstract: We generalize Adleman's algorithm for computing discrete logarithms over prime finite fields to an arbitrary finite field. The algorithm works in subexponential time provided some assumption is true. This assumption is supported by experimental results presented here. Let Fq be a finite field with q elements. The multiplicative group F* of the field is cyclic of order q 1. The discrete logarithm problem consists in the fast computation of χ such that a* = b, where a, 6 e F*. The importance of this problem for mathematics is related to its applications in cryptology [1]. A method based on the structure of the group F* was first suggested in [2]. Its complexity is exponential if q 1 has a factor of an exponential size. In 1979 Adleman has shown [3] that, if q is a prime number, one can find the logarithms over Fq in time exp(c(log?loglog oo and c is a small constant. We take a quick look at this method. A homomorphism Ζ -> Fq of the ring of integers Ζ into the prime finite field Fq is used in the algorithm. Let qt be the t first primes, where t is a parameter of the method which consists of three steps. (1) We generate a random integer m, l < m < q 2, and find the residue οι ΞΞ a (mod q) . Thus, &i is an integer from 1 to q 1. If the maximum prime factor of 61 is greater than qt, then we generate another random ra. Let, finally, 61 = ̂ ...?i'. Then we obtain linear equation modulo q — 1: l\\x\\ + 12X2 + . . . + ItXt = m (mod q 1) with respect to the unknown logarithms Xi of the elements

Three-Dimensional Viscous Flows Between Concentric Cylinders Executing Axially Variable Oscillations: A Hybrid Spectral/Finite Difference Solution

Abstract: A hybrid spectral/finite difference method is developed for the analysis of three-dimensional unsteady viscous flows between concentric cylinders subjected to fully developed laminar flow and executing transverse oscillations. This method uses a partial spectral collocation approach, based on spectral expansions of the flow parameters in the transverse coordinates and time, in conjunction with a finite difference discretization of the axial derivatives. The finite difference discretization uses central differencing for the diffusion derivatives and a mixed central-upwind differencing for the convective derivatives, in terms of the local mesh Reynolds number. This mixed scheme can be used with coarser as well as finer axial mesh spacings, enhancing the computational efficiency. The hybrid spectral/finite difference method efficiently reduces the problem to a block-tridiagonal matrix inversion, avoiding the numerical difficulties otherwise encountered in a complete three-dimensional spectral-collocation approach. This method is used to compute the unsteady fluid-dynamic forces, the real and imaginary parts of which are related, respectively, to the added-mass and viscous-damping coefficients.

A finite difference method for a boundary value problem related to the Kolmogorov equation

Abstract: We propose an explicit and an implicit difference scheme for a boundary value problem related to the Kolmogorov equation. Stability conditions and convergence results are given.

Summation by parts formula for noncentered finite differences

Abstract: Boundary conditions for general high order nite di erence approxima tions to d dx are constructed It is proved that it is always possible to nd boundary conditions and a weighted norm in such a way that a summation by parts formula holds

Field-to-wire coupling using the finite element-time domain (FE-TD) method

Abstract: A new numerical method is proposed to analyze transient induced effects in thin wires excited by external fields. Maxwell's curl equations and transmission line equations are solved by the point-matching finite element-time domain (FE-TD) method. The method is based on a finite element discretization of space and a finite difference discretization of time. The leap-frog scheme is adopted and an explicit solution scheme is obtained. >

Optimized locally exact numerical scheme based on finite variable difference method and characteristic polynomial analysis method: in case of convection-diffusion equation without source terms

Abstract: A new method of Finite Variable Difference Method (FVDM) is presented The feature of this method exists in a procedure to determine the finite spatial difference, in which the total deviation of the numerical solution from the exact solution is minimized, under the condition that roots of the resulting characteristic equation are always non-negative to insure numerical stability The optimum spatial difference of the LECUSSO scheme for the linear convection-diffusion equation is numerically derived in terms of mesh Reynolds numbers This optimization highly improves the numerical accuracy of the LECUSSO scheme for linear convection-diffusion equations without numerical oscillations at sufficiently large mesh Reynolds numbers of up to 1,000

Nonlinear difference approximations for evolutionary PDEs

Abstract: The authors describe a procedure to improve both the accuracy and computational efficiency of finite difference schemes used to simulate nonlinear PDEs. The underlying idea is that of enslaving, which is the estimation of the small unresolved scales in terms of the larger resolved scales. They discuss details of the procedure and illustrate them in the context of the forced Burgers` equation in one dimension. They present computational examples that demonstrate the predicted increases in accuracy and efficiency.

Parallel finite difference solution of general inhomogeneous anisotropic bio-electrostatic problems

Abstract: Finite difference solution of bio-electrostatic problems have been limited mainly to systems where the conductivity is orthotropic, i.e., a strictly diagonal conductivity tensor. This in turn has limited the use of the finite difference technique in modeling the detailed structure of various muscles, where the fibers can be in arbitrary directions and an accurate representation demands a non-diagonal conductivity tensor. Here, the authors describe a new three-dimensional finite difference formulation that is valid for solving the governing Laplace equation in structures with an inhomogeneous and non-diagonal conductivity tensor. In addition, a data parallel computer is used in the finite difference implementation to provide the memory and reduction in solution time for solving large problems. The finite difference algorithm will be described together with its parallel implementation, and numerical results is presented.

Optimized Locally Exact Numerical Scheme Based on Finite Variable Difference Method and Characteristic Polynomial Analysis Method

Abstract: A new method of Finite Variable Difference Method (FVDM) is presented. The feature of this method exists in a procedure to determine the finite spatial difference, in which the total deviation of the numerical solution from the exact solution is minimized, under the condition that roots of the resulting characteristic equation are always non-negative to insure numerical stability. The optimum spatial difference of the LECUSSO scheme for the linear convection-diffusion equation is numerically derived in terms of mesh Reynolds numbers. This optimization highly improves the numerical accuracy of the LECUSSO scheme for linear convection-diffusion equations without numerical oscillations at sufficiently large mesh Reynolds numbers of up to 1,000.

Uniqueness of finite difference approximations to elliptic systems of partial differential equations

Abstract: We consider regular finite difference discretizations of systems of elliptic partial differential equations. Regularity of the partial differential system and of the finite difference approximation is assumed. Uniqueness of the finite difference solution is proved in the case that the differential system has a unique solution. In the case of a non-unique solution, the dimension of the finite difference operator kernel is shown to be no larger than the dimension of the corresponding differential operator kernel.

Wavelets as a numerical tool

Abstract: The stability of a plane, premixed, adiabatic flame is re-examined for finite activation energies. Preliminary results are given for one-dimensional disturbances. The full governing equations for an unsteady propagating flame are then solved using a novel numerical approach called WOFD, a waveleto-ptimized finite difference method. This method is briefly described here and preliminary results are reported. As a result of this work, we strongly recommend that new and old algorithms should be tested at activation energies of 40 and 60 and compared to the results presented here as a gauge to the accuracy of the numerical solution.