Topic
Finite difference method
About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.
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TL;DR: A stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon and inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation.
Abstract: We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is \(O( (\Delta x)^2 + (\Delta t)^2)\). Numerical examples demonstrate the effectiveness of the proposed scheme.
245 citations
TL;DR: A local-in-time error estimate is presented that ensures the pointwise convergence of the scheme, which is based on a convex splitting of a discrete pseudoenergy and is semi-implicit.
Abstract: We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.
243 citations
TL;DR: In this paper, a novel numerical technique is proposed for the electromagnetic characterization of the scattering by a three-dimensional cavity-backed aperture in an infinite ground plane, combining the finite element and boundary integral methods to formulate a system of equations for the solution of the aperture fields and those inside the cavity.
Abstract: A novel numerical technique is proposed for the electromagnetic characterization of the scattering by a three-dimensional cavity-backed aperture in an infinite ground plane. The technique combines the finite element and boundary integral methods to formulate a system of equations for the solution of the aperture fields and those inside the cavity. Specifically, the finite element method is used to formulate the fields in the cavity region, and the boundary integral approach is used in conjunction with the equivalence principle to represent the fields above the ground plane. Unlike traditional approaches, the proposed technique does not require a knowledge of the cavity's Green's function and is, therefore, applicable to arbitrary shape depressions and material fillings. Furthermore, the proposed formulation leads to a system having a partly full and partly sparse as well as symmetric and banded matrix which can be solved efficiently using special algorithms. >
243 citations
TL;DR: In this paper, two antennas are considered, a cylindrical monopole and a conical monopole, which are driven through an image plane from a coaxial transmission line and analyzed by a straightforward application of the finite-difference-time-domain (FD-TD) method.
Abstract: Two antennas are considered, a cylindrical monopole and a conical monopole. Both are driven through an image plane from a coaxial transmission line. Each of these antennas corresponds to a well-posed theoretical electromagnetic boundary value problem and a realizable experimental model. These antennas are analyzed by a straightforward application of the finite-difference-time-domain (FD-TD) method. The computed results for these antennas are shown to be in excellent agreement with accurate experimental measurements for both the time domain and the frequency domain. The graphical displays presented for the transient near-zone and far-zone radiation from these antennas provide physical insight into the radiation process. >
242 citations
01 Oct 2000
TL;DR: A new multiscale method in surface processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems to smooth discretized surfaces while simultaneously enhancing geometric features such as edges and corners.
Abstract: A new multiscale method in surface processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems. Its aim is to smooth discretized surfaces while simultaneously enhancing geometric features such as edges and corners. This is obtained by an anisotropic curvature evolution, where time is the multiscale parameter. Here, the diffusion tensor depends on the shape operator of the evolving surface. A spatial finite element discretization on arbitrary unstructured triangular meshes and a semi-implicit finite difference discretization in time are the building blocks of the easy to code algorithm presented. The systems of linear equations in each timestep are solved by appropriate, preconditioned iterative solvers. Different applications underline the efficiency and flexibility of the presented type of surface processing tool.
242 citations