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: This paper shows that a classic optical flow technique by Nagel and Enkelmann can be regarded as an early anisotropic diffusion method with a diffusion tensor, and introduces three improvements into the model formulation that avoid inconsistencies caused by centering the brightness term and the smoothness term in different images.
Abstract: In this paper we show that a classic optical flow technique by Nagel and Enkelmann (1986, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 8, pp. 565–593) can be regarded as an early anisotropic diffusion method with a diffusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and the smoothness term in different images, (ii) use a linear scale-space focusing strategy from coarse to fine scales for avoiding convergence to physically irrelevant local minima, and (iii) create an energy functional that is invariant under linear brightness changes. Applying a gradient descent method to the resulting energy functional leads to a system of diffusion–reaction equations. We prove that this system has a unique solution under realistic assumptions on the initial data, and we present an efficient linear implicit numerical scheme in detail. Our method creates flow fields with 100 % density over the entire image domain, it is robust under a large range of parameter variations, and it can recover displacement fields that are far beyond the typical one-pixel limits which are characteristic for many differential methods for determining optical flow. We show that it performs better than the optical flow methods with 100 % density that are evaluated by Barron et al. (1994, Int. J. Comput. Vision, Vol. 12, pp. 43–47). Our software is available from the Internet.
344 citations
TL;DR: In contrast with earlier nodal simulators, more recent nodal diffusion methods are characterized by the systematic derivation of spatial coupling relationships that are entirely consistent with the multigroup diffusion equation as discussed by the authors, which most often are derived by developing approximations to the one-dimensional equations obtained by integrating the multidimensional diffusion equation over directions transverse to each coordinate axis.
344 citations
TL;DR: This work presents a high-order modified immersed interface method for the 2D, unsteady, incompressible Navier-Stokes equations in stream function-vorticity formulation that employs an explicit fourth-order Runge-Kutta time integration scheme, and a nine-point, four-order compact discretization of the Poisson equation for computation of the stream function.
343 citations
TL;DR: In this article, a simple method of constructing adaptive grids is presented and the benefits are demonstrated by calculations of Sod's shock tube problem (G. A. Sod, J. Comput. Phys. 27, 1 (1978)) and of a supernova explosion.
341 citations
TL;DR: This numerical method combines the alternating directions implicit (ADI) approach with a Crank-Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method.
339 citations