Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solution with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.

New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry

Abstract: Traditional resistivity tools are designed to function in vertical wells. In horizontal well environments, the interpretation of resistivity logs becomes much more difficult because of the nature of 3-D effects such as highly deviated bed boundaries and invasion. The ability to model these 3-D effects numerically can greatly facilitate the understanding of tool response in different formation geometries. Three-dimensional modeling of induction tools requires solving Maxwell's equations in a discrete setting, either finite element or finite difference. The solutions of resulting discretized equations are computationally expensive, typically on the order of 30 to 60 minutes per log point on a workstation. This is unacceptable if the 3-D modeling code is to be used in interpreting induction logs. In this paper we propose a new approach for solutions to Maxwell's equations. The new method is based on the spectral Lanczos decomposition method (SLDM) with Krylov subspaces generated from the inverse powers of the Maxwell operator. This new approach significantly speeds up the convergence of standard SLDM for the solution of Maxwell's equations while retaining the advantages of standard SLDM such as the ability of solving for multiple frequencies and eliminate completely spurious modes. The cost of evaluating powers of the matrix inverse of the stiffness operator is effectively equivalent to the cost of solving a scalar Poisson's equation. This is achieved by a decomposition of the stiffness operator into the curl-free and divergence-free projections. The solution of the projections can be computed by discrete Fourier transforms (DFT) and preconditioned conjugate gradient iterations. The convergence rate of the new method improves as frequency decreases, which makes it more attractive for low-frequency applications. We apply the new solution technique to model induction logging in geophysical prospecting applications, giving rise to two orders of magnitude convergence improvement over the standard Krylov subspace approach and more than an order of magnitude speed-up in terms of overall execution. This makes it feasible to routinely use 3-D modeling for model-based interpretation, a break-through in induction logging and interpretation.

Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems

Abstract: The response of a structural system to white noise excitation (deltacorrelated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by standard sequential finite element method (FEM) using C0 shape function and Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by Spencer, Bergman and Wojtkiewicz is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed.