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Difference Approximation (difference + approximation)

Distribution by Scientific Domains
Engineering64%
Mathematics and Statistics28%

Kinds of Difference Approximation

  • finite difference approximation
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    Selected Abstracts

    Calculation of the wave propagation angle in complex media: application to turning wave simulations

    GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2009
    Xiaofeng Jia
    SUMMARY The wave propagation angle is one of the key factors in seismic processing methods. For the dual-domain propagators, it is sometimes necessary to acquire the wave propagation angle in the space-frequency domain instead of the wavenumber domain or the angle domain. We propose a method dealing with this problem, in which the wavefield gradient is used for the calculation of the wave propagation angle. The wavefield gradient can be directly obtained by either the finite difference approximation or the marching expression of the propagator. This method is not applicable in the case of extremely low frequency due to the comparability between the wavelength and the grid interval. Combined with the superwide-angle one-way propagator, this approach is instrumental in simulating the turning wave, which is hard to be handled by the traditional one-way propagator. Numerical examples show the good performance of the superwide-angle one-way propagator with our approach involved. The turning wave is modelled accurately; as a result, a high-quality image of the overhanging salt flank can be obtained. [source]

    An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2008
    Lourenço Beirão da Veiga
    Abstract We present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form. We show that this estimator is reliable and efficient with respect to an energy-type error comprising both flux and pressure. Its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes, different permeability tensors, and different types of computational meshes. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Towards very high-order accurate schemes for unsteady convection problems on unstructured meshes

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8-9 2005
    R. Abgrall
    Abstract We construct several high-order residual-distribution methods for two-dimensional unsteady scalar advection on triangular unstructured meshes. For the first class of methods, we interpolate the solution in the space,time element. We start by calculating the first-order node residuals, then we calculate the high-order cell residual, and modify the first-order residuals to obtain high accuracy. For the second class of methods, we interpolate the solution in space only, and use high-order finite difference approximation for the time derivative. In doing so, we arrive at a multistep residual-distribution scheme. We illustrate the performance of both methods on several standard test problems. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    Reduced order state-space models from the pulse responses of a linearized CFD scheme

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2003
    Ann L. Gaitonde
    This paper describes a method for obtaining a time continuous reduced order model (ROM) from a system of time continuous linear differential equations. These equations are first put into a time discrete form using a finite difference approximation. The unit sample responses of the discrete system are calculated for each system input and these provide the Markov parameters of the system. An eigenvalue realization algorithm (ERA) is used to construct a discrete ROM. This ROM is then used to obtain a continuous ROM of the original continuous system. The focus of this paper is on the application of this method to the calculation of unsteady flows using the linearized Euler equations on moving meshes for aerofoils undergoing heave or linearized pitch motions. Applying a standard cell-centre spatial discretization and taking account of mesh movement a continuous system of differential equations is obtained which are continuous in time. These are put into discrete time form using an implicit finite difference approximation. Results are presented demonstrating the efficiency of the system reduction method for this system. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Analysis of super compact finite difference method and application to simulation of vortex,shock interaction

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2001
    Fu Dexun
    Abstract Turbulence and aeroacoustic noise high-order accurate schemes are required, and preferred, for solving complex flow fields with multi-scale structures. In this paper a super compact finite difference method (SCFDM) is presented, the accuracy is analysed and the method is compared with a sixth-order traditional and compact finite difference approximation. The comparison shows that the sixth-order accurate super compact method has higher resolving efficiency. The sixth-order super compact method, with a three-stage Runge,Kutta method for approximation of the compressible Navier,Stokes equations, is used to solve the complex flow structures induced by vortex,shock interactions. The basic nature of the near-field sound generated by interaction is studied. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    Numerical nonlinear observers using pseudo-Newton-type solvers

    INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 17 2008
    Shigeru HanbaArticle first published online: 12 DEC 200
    Abstract In constructing a globally convergent numerical nonlinear observer of Newton-type for a continuous-time nonlinear system, a globally convergent nonlinear equation solver with a guaranteed rate of convergence is necessary. In particular, the solver should be Jacobian free, because an analytic form of the state transition map of the nonlinear system is generally unavailable. In this paper, two Jacobian-free nonlinear equation solvers of pseudo-Newton type that fulfill these requirements are proposed. One of them is based on the finite difference approximation of the Jacobian with variable step size together with the line search. The other uses a similar idea, but the estimate of the Jacobian is mostly updated through a BFGS-type law. Then, by using these solvers, globally stable numerical nonlinear observers are constructed. Numerical results are included to illustrate the effectiveness of the proposed methods. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Jacobi,Davidson methods for cubic eigenvalue problems

    NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2005
    Tsung-Min Hwang
    Abstract Several Jacobi,Davidson type methods are proposed for computing interior eigenpairs of large-scale cubic eigenvalue problems. To successively compute the eigenpairs, a novel explicit non-equivalence deflation method with low-rank updates is developed and analysed. Various techniques such as locking, search direction transformation, restarting, and preconditioning are incorporated into the methods to improve stability and efficiency. A semiconductor quantum dot model is given as an example to illustrate the cubic nature of the eigenvalue system resulting from the finite difference approximation. Numerical results of this model are given to demonstrate the convergence and effectiveness of the methods. Comparison results are also provided to indicate advantages and disadvantages among the various methods. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    The Poisson equation with local nonregular similarities

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2001
    Alexander Yakhot
    Abstract Moffatt and Duffy [1] have shown that the solution to the Poisson equation, defined on rectangular domains, includes a local similarity term of the form: r2log(r)cos(2,). The latter means that the second (and higher) derivative of the solution with respect to r is singular at r = 0. Standard high-order numerical schemes require the existence of high-order derivatives of the solution. Thus, for the case considered by Moffatt and Duffy, the high-order finite-difference schemes loose their high-order convergence due to the nonregularity at r = 0. In this article, a simple method is outlined to regain the high-order accuracy of these schemes, without the need of any modification in the scheme's algorithm. This is a significant consideration when one wants to use a given finite-difference computer code for problems with local nonregular similarity solutions. Numerical examples using the modified scheme in conjunction with a sixth-order finite difference approximation are provided. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:336,346, 2001 [source]

    Numerical methods for fourth-order nonlinear elliptic boundary value problems

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2001
    C. V. Pao
    Abstract The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth-order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence-comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two-point boundary-value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347,368, 2001 [source]

    A continuum sensitivity method for finite thermo-inelastic deformations with applications to the design of hot forming processes

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002
    Shankar Ganapathysubramanian
    Abstract A computational framework is presented to evaluate the shape as well as non-shape (parameter) sensitivity of finite thermo-inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo-mechanical deformation of hyperelastic thermo-viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite-dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Numerical solutions of fully non-linear and highly dispersive Boussinesq equations in two horizontal dimensions

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2004
    David R. Fuhrman
    Abstract This paper investigates preconditioned iterative techniques for finite difference solutions of a high-order Boussinesq method for modelling water waves in two horizontal dimensions. The Boussinesq method solves simultaneously for all three components of velocity at an arbitrary z -level, removing any practical limitations based on the relative water depth. High-order finite difference approximations are shown to be more efficient than low-order approximations (for a given accuracy), despite the additional overhead. The resultant system of equations requires that a sparse, unsymmetric, and often ill-conditioned matrix be solved at each stage evaluation within a simulation. Various preconditioning strategies are investigated, including full factorizations of the linearized matrix, ILU factorizations, a matrix-free (Fourier space) method, and an approximate Schur complement approach. A detailed comparison of the methods is given for both rotational and irrotational formulations, and the strengths and limitations of each are discussed. Mesh-independent convergence is demonstrated with many of the preconditioners for solutions of the irrotational formulation, and solutions using the Fourier space and approximate Schur complement preconditioners are shown to require an overall computational effort that scales linearly with problem size (for large problems). Calculations on a variable depth problem are also compared to experimental data, highlighting the accuracy of the model. Through combined physical and mathematical insight effective preconditioned iterative solutions are achieved for the full physical application range of the model. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Boundary element analysis of driven cavity flow for low and moderate Reynolds numbers

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2001
    M. Aydin
    Abstract A boundary element method for steady two-dimensional low-to-moderate-Reynolds number flows of incompressible fluids, using primitive variables, is presented. The velocity gradients in the Navier,Stokes equations are evaluated using the alternatives of upwind and central finite difference approximations, and derivatives of finite element shape functions. A direct iterative scheme is used to cope with the non-linear character of the integral equations. In order to achieve convergence, an underrelaxation technique is employed at relatively high Reynolds numbers. Driven cavity flow in a square domain is considered to validate the proposed method by comparison with other published data. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2005
    Liliana Borcea
    We consider finite difference approximations of solutions of inverse Sturm-Liouville problems in bounded intervals. Using three-point finite difference schemes, we discretize the equations on so-called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm-Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand-Levitan formulation. © 2005 Wiley Periodicals, Inc. [source]