Showing papers on "Finite difference coefficient published in 1991"
TL;DR: In this paper, a connection between the concepts of determining nodes and inertial manifolds with that of finite difference and finite volumes approximations to dissipative partial differential equations is presented.
Abstract: The authors present a connection between the concepts of determining nodes and inertial manifolds with that of finite difference and finite volumes approximations to dissipative partial differential equations. In order to illustrate this connection they consider the 1D Kuramoto-Sivashinsky equation as a instructive paradigm. They remark that the results presented here apply to many other equations such as the 1D complex Ginzburg-Landau equation, the Chafee-Infante equation, etc.
150 citations
TL;DR: In this paper, three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered, based on the forward Euler, the backward Euler and the Crank-Nicolson methods.
Abstract: Three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered. The schemes are based on the forward Euler, the backward Euler and the Crank-Nicolson methods. Error estimates are derived in maximum norm. Results from a numerical experiment are presented.
105 citations
TL;DR: The individua band profiles obtained for binary mixtures are more accurate than those derived using one of several possible finite difference methods, and the advantage of the better accuracy is compensated by a considerably higher computation time.
74 citations
TL;DR: In this paper, the authors show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations (PDES) can mean the difference between dissipation and blow up.
Abstract: The authors show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations (PDES) can mean the difference between dissipation and blow up. For fixed parameter values and arbitrarily fine discretizations they construct solutions which blow up in finite time for two semi-discrete schemes. They also show the existence of spurious steady states whose unstable manifolds, in some cases, contain solutions which explode. This connection between the blow-up phenomenon and spurious steady states is also explored for Galerkin and nonlinear Galerkin semi-discrete approximations. Two fully discrete finite difference schemes derived from a third semi-discrete scheme, reported to be dissipative, are analysed. Both latter schemes are shown to have a stability condition which is independent of the initial data.
73 citations
TL;DR: The Laplace transform finite difference (LTFD) method as discussed by the authors was developed to solve the partial differential equation (PDE) of transient flow through porous media, which is semianalytical in time and numerical in space by solving the discretized PDE in the Laplace space.
Abstract: A new numerical method, the Laplace transform finite difference (LTFD) method, was developed to solve the partial differential equation (PDE) of transient flow through porous media. LTFD provides a solution which is semianalytical in time and numerical in space by solving the discretized PDE in the Laplace space and numerically inverting the transformed solution vector. The effects of the traditional treatment of the time derivative on accuracy and stability are rendered irrelevant because time is no longer a consideration. For a single time step, LTFD requires no more than eight matrix solutions and an execution time eight times longer than the analogous finite difference (FD) requirement without an increase in storage. This disadvantage is outweighed by an unlimited time step size without any loss of accuracy, a superior accuracy, and a stable, nonincreasing round off error. Thus, a problem in standard FD format may require several hundred time steps and matrix inversions between the initial condition and the desired solution time, but LTFD requires only one time step and no more than eight matrix inversions to achieve a more accurate result.
58 citations
35 citations
TL;DR: In this paper, the authors compared seven methods for calculating sensitivity derivatives for a two-material beam problem with a moving interface and found that as the mesh is refined, displacement sensitivity derivatives converge more slowly than the displacements.
Abstract: The calculation of sensitivity of the response of a structure modeled by finite elements to shape variation is known to be subject to numerical difficulties. The accuracy of a given method is typically measured against the yard stick of finite-difference sensitivity calculation. The present paper demonstrates with a simple example that this approach may be flawed because of discretization errors associated with the finite element mesh. Seven methods for calculating sensitivity derivatives are compared for a two-material beam problem with a moving interface. It is found that as the mesh is refined, displacement sensitivity derivatives converge more slowly than the displacements. Six of the methods agree fairly well, but the adjoint variational surface method provides substantially different results. However, the difference is found to reflect convergence from another direction to the same answer rather than reduced accuracy. Additionally, it is observed that small derivatives are particularly prone to accuracy problems.
26 citations
TL;DR: In this paper, it was shown that this simple idea does not generalize to multidimensional elliptic problems except when the coefficients are periodic, where the coefficients can be périodiques.
Abstract: — It is possible to solve numericaïly problems with rapidly oscillating coefficients (homogenization) without resolving the rapid oscillations ? In some cases, in one dimensional problems and in some multidimensional hyperbolic problems, this can be done by using grids that are irregularly spaced relative to the rapid oscillations. In this paper we show that this simple idea does not generalize to multidimensional elliptic problems except when the coefficients are periodic. Résumé. — Peut-on résoudre numériquement des équations avec des coefficients à variation rapide (homogénéisation) sans un échantillonnage détaillé des échelles les plus fines ? Pour certains problèmes monodimensionnels, ainsi que pour certaines équations hyperboliques, cela peut se faire en utilisant des maïllages placés de façon irreguliere par rapport aux échelles d'oscillation rapide. Dans cette note, nous montrons que cette méthode simple ne peut se généraliser aux problèmes elliptiques multidimensionnels, à l'exception du cas où les coefficients sont périodiques.
25 citations
01 Apr 1991
TL;DR: In this article, a non-uniform finite difference method is presented for an analysis of arbitrarily-shaped optical waveguides, where preconditioning of discretised coefficient matrices by similarity transformations is used to improve the accuracy of the solution.
Abstract: A nonuniform finite difference method is presented for an analysis of arbitrarily-shaped optical waveguides. By introducing preconditioning of discretised coefficient matrices by similarity transformations, it has become possible to improve the accuracy of the solution without paying any penalty in terms of computing time. From the comparison between the exact analytical method and the nonuniform finite difference method for an analysis of planar slab waveguides, it has been clarified that the mesh refinement near the material interfaces plays a dominant role in determining the accuracy of solutions. Furthermore, the nonuniform finite difference method is used to model buried-channel waveguides and is compared with the effective-index method. >
15 citations
TL;DR: In this paper, conditions for the continuity and differentiability of solutions of initial value problems and boundary value problems for the n th order finite difference equation, u(m + n) /(-, u(.), u(. + 1) u(, + n- 1)),- Z.
Abstract: Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the n th order finite difference equation, u(m + n) /(-, u(.), u(. + 1) u(, + n- 1)),- Z.
15 citations
TL;DR: In this article, the authors extended the concept of the elliptic projection to the finite difference case and proved second-order error estimates in space and time in the 12 norm for the heat equation, the Schrodinger equation, and the wave equation.
Abstract: We analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation, the Schrodinger equation, and the wave equation by finite difference methods. Extending the concept of the elliptic projection, well known from the analysis of Galerkin finite element methods, to our finite difference case, we prove second-order error estimates in space and time in the 12 norm.
TL;DR: In this article, a finite difference technique has been developed to study the Newtonian jet swell problem, where the stream function and vorticity were used as dependent variables to describe the jet flow.
Abstract: A finite difference technique has been developed to study the Newtonian jet swell problem. The streamfunction and vorticity were used as dependent variables to describe the jet flow. The boundary-fitted co-ordinate transformation method was adopted to map the flow geometry into a rectangular domain. The standard finite difference method was then applied for solving the flow equations. The location of the jet free surface was updated by the kinematic boundary condition, and an adjustable parameter was included in the free-surface iteration. We could obtain numerical solutions for the Reynolds number as high as 100, and the differences between the present study and previous finite element simulations on the jet swell ratio are less than 5%.
TL;DR: The problems of model initialization, mathematical stability and dispersion, as well as the usefulness of the local character of these techniques for parallel computing as opposed to the global nature of Fourier techniques and the resulting implications for data motion are discussed.
Abstract: The treatment of the full acoustic wave equation by second order finite differences in space and time has been successfully used in exploration geophysics over the past one and a half decades for forward modeling. Migration, the inverse of modeling, was traditionally done by depth extrapolation of the upcoming wavefield. However, extrapolation in depth is not always a stable process. Furthermore, in treating only the upcoming energy, one must either sacrifice the generality of the full wave equation by using the 15 or 45 degree finite difference operator in space and time, or one must resort to Fourier methods such as the phase shift technique in which severe limitations are imposed on the velocity function. It is known that higher order finite difference methods significantly reduce the numerical dispersion associated with such techniques. However, efforts to apply this result to modeling problems, and not to mention inverse problems, have only recently been published. Here we investigate the application of fourth and higher order difference methods to the full acoustic wave equation for both modeling and reverse time migration. We also discuss the problems of model initialization, mathematical stability and dispersion, as well as the usefulness of the local character of these techniques for parallel computing as opposed to the global nature of Fourier techniques and the resulting implications for data motion, especially in 3D. Finally, we show an example of reverse time migration applied to a real data set from Western Australia.
TL;DR: In this paper, the smoothed solution against which the finite element stresses are compared is based on a reformulation of the finite difference method that permits irregular meshes and complex boundary conditions to be analyzed.
Abstract: the smoothed solution against which the finite element stresses are compared is based on a reformulation of the finite difference method that permits irregular meshes and complex boundary conditions to be analyzed
TL;DR: In this article, a correlated quasiparticle approximation, based on the inclusion of diagonal two-body terms in the exponent of the statistical operator, is discussed, which is seen to yield a substantial improvement over the conventional finite temperature mean-field approach for finite systems.
01 Dec 1991
TL;DR: The authors present a parallel implementation of the finite difference time domain (FDTD) method using part of the Fortran 8x CM-2 code and results are given for its application to electromagnetic scattering from an Al cube.
Abstract: The authors present a parallel implementation of the finite difference time domain (FDTD) method. Part of the Fortran 8x CM-2 code is given. Results are given for its application to electromagnetic scattering from an Al cube. >
TL;DR: In this article, a discussion of a numerical inaccuracy that may occur when a staggered finite difference grid is used to solve the finite differenced multiphase flow equations is presented.
TL;DR: The new, slightly modified algorithm, when used with explicit (forward time-difference) modelling, gave analytically exact results for the charge interaction part of a similar trial calculation.
Abstract: This note describes a minor modification to the recently published algorithm 'Q-COUPLE' designed for adding charge-charge interactions between diffusing species to time-dependent one-dimensional finite difference diffusion calculations. The original proposal concerned a simple way of doing this for Crank-Nicolson central time-difference schemes, and gave useful, but only approximate, agreement with theory in tests where the charge-coupling could also be calculated analytically. The new, slightly modified algorithm, when used with explicit (forward time-difference) modelling, gave analytically exact results for the charge interaction part of a similar trial calculation.
07 Jan 1991
01 Jun 1991
TL;DR: An algorithm to generate low complexity normal bases in finite fields is presented and produces the multiplication matrix for the normal basis multiplication of GF'(qn) : GI’(q) in 0(n2 log2 n log q) bit–operations.
Abstract: An algorithm to generate low complexity normal bases in finite fields is presented. This algorithm generalizes the method of Ash et al. to fields of arbitrary characteristic. It can be applied to most finite fields and produces (under certain conditions) the multiplication matrix for the normal basis multiplication of GF’(qn) : GI’(q) in 0(n2 log2 n log q) bit–operations.
TL;DR: In this paper, two domain-adaptive finite difference methods are presented and applied to study the dynamic response of incompressible, inviscid, axisymmetric liquid membranes subject to imposed sinusoidal pressure oscillations.
Abstract: SUMMARY Two domain-adaptive finite difference methods are presented and applied to study the dynamic response of incompressible, inviscid, axisymmetric liquid membranes subject to imposed sinusoidal pressure oscillations. Both finite difference methods map the time-dependent physical domain whose downstream boundary is unknown onto a fixed computational domain. The location of the unknown time-dependent downstream boundary of the physical domain is determined from the continuity equation and results in an integrodifferential equation which is non-linearly coupled with the partial differential equations which govern the conservation of mass and linear momentum and the radius of the liquid membrane. One of the finite difference methods solves the non-conservative form of the governing equations by means of a block implicit iterative method. This method possesses the property that the Jacobian matrix of the convection fluxes has an eigenvalue of algebraic multiplicity equal to four and of geometric multiplicity equal to one. The second finite difference procedure also uses a block implicit iterative method, but the governing equations are written in conservation law form and contain an axial velocity which is the difference between the physical axial velocity and the grid speed. It is shown that these methods yield almost identical results and are more accurate than the non-adaptive techniques presented in Part I. It is also shown that the actual value of the pressure coefficient determined from linear analyses can be exceeded without affecting the stability and convergence of liquid membranes if the liquid membranes are subjected to sinusoidal pressure variations of sufficiently high frequencies.
01 Jan 1991
TL;DR: In this paper, a three-dimensional treatment of transmission line discontinuity problems by the finite difference method is presented, which allows to compute the scattering parameters of an arbitrary structure, including coupling of higher order modes.
Abstract: A three-dimensional treatment of transmission line discontinuity problems by the finite difference method is presented. Maxwell’s equations are solved in the frequency domain by solution of a boundary value problem. The presented method allows to compute the scattering parameters of an arbitrary structure, including coupling of higher order modes. The general formulation and the procedure of the method is described. Verification calculations are given and results for different microstrip discontinuities are included for the lossy and non-lossy case.
TL;DR: In this article, an efficient finite difference scheme for the inviscid terms of the three-dimensional, compressible flow equations for chemical non-equilibrium gases is presented, which represents an extension and an improvement of one proposed by the author.
Abstract: An efficient finite difference scheme is presented for the inviscid terms of the three-dimensional, compressible flow equations for chemical non-equilibrium gases. This scheme represents an extension and an improvement of one proposed by the author, and includes operator splitting.
TL;DR: In this paper, the combined scheme of finite elements and finite differences is applied to axisymmetric exterior field problems, as studied by Wong and Ciric, and the results of some calculations are compared with the exact solutions where possible.
Abstract: The combined scheme of finite elements and finite differences, which was developed by Yano and already successfully applied to the two-dimensional Laplace equation, is applied to axisymmetric exterior-field problems, as studied by Wong and Ciric. In order to illustrate the validity of the proposed technique, the results of some calculations are compared with the exact solutions where possible.
TL;DR: In this article, a second-order Godunov-type finite difference scheme is applied to the governing equations of finite amplitude plane motion of a hyperelastic string, expressed in conservation form, are a hyperbolic system of four first-order partial differential equations.
Abstract: A second-order Godunov-type finite difference scheme is applied to the governing equations of finite amplitude plane motion of a hyperelastic string. The governing equations, expressed in conservation form, are a hyperbolic system of four first-order partial differential equations. Numerical results are presented for a string subjected to a transverse impact at its midpoint, and the string is modelled by a strain energy function which is realistic for simple tension. Similarity solutions, valid for times before reflections occur, are in excellent agreement with the results from the finite difference scheme and provide a partial check on the validity of the scheme. Numerical results, which include the effects of reflections and interactions of waves, are shown graphically.
TL;DR: In this paper, a new method for solving the Navier-Stokes equations in primitive variable form is presented, which uses finite difference operators for the convective terms within the framework of a penalty finite element method.
TL;DR: In this article, a dynamic analysis of waveguide structures containing dielectric and metal strips is presented, where the analysis utilizes a finite difference frequency domain procedure to reduce the problem to a symmetric matrix eigenvalue problem.
Abstract: Dynamic analysis of waveguide structures containing dielectric and metal strips is presented. The analysis utilizes a finite difference frequency domain procedure to reduce the problem to a symmetric matrix eigenvalue problem. Since the matrix is also sparse, the eigenvalue problem can be solved quickly and efficiently using the conjugate gradient method resulting in considerable savings in computer storage and time. Comparison is made with the analytical solution for the loaded dielectric waveguide case. For the microstrip case, we get both waveguide modes and quasi-TEM modes. The quasi-TEM modes in the limit of zero frequency are checked with the static analysis which also uses finite difference. Some of the quasi-TEM modes are spurious. This article describes their origin and discusses how to eliminate them. Numerical results are presented to illustrate the principles.
01 Jan 1991
TL;DR: In this article, a new method is presented for the calculation of shape design sensitivity with kinematical design boundary, which modifies the traditional finite difference approach such that the variation of the structural response due to the change of the kinematic boundary is replaced by an equivalent problem, and the final design sensitivity is expressed as the solutions of the initial structure under the perturbation displacements on the design boundary.
Abstract: A new method is presented in this paper for the calculation of shape design sensitivity with kinematical design boundary. This new method modifies the traditional finite difference approach, such that the variation of the structural response due to the change of the kinematic boundary is replaced by an equivalent problem, and the final design sensitivity is expressed as the solutions of the initial structure under the perturbation displacements on the design boundary. Two examples are used to demonstrate the proposed new formulation.