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: Finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions are analyzed and error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts are derived.
Abstract: We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h2 + τ2) in the l2-norm and discrete H1-norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the l∞-norm by using the inverse inequality and the l2-norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.
159 citations
TL;DR: This article proposes to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduces fast and robust solvers and reduces the memory consumption of the Moulinec–Suquet algorithms by 50%.
Abstract: Summary
In this article, we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers. Our method shares some properties with the FFT-based homogenization technique of Moulinec and Suquet, which has received widespread attention recently because of its robustness and computational speed. These similarities include the use of FFT and the resulting performing solvers. The staggered grid discretization, however, offers three crucial improvements. Firstly, solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by Moulinec–Suquet's discretization. Secondly, the iteration numbers of our solvers are bounded independently of the grid size and the contrast. In particular, our solvers converge for three-dimensional porous structures, which cannot be handled by Moulinec–Suquet's method. Thirdly, the finite difference discretization allows for algorithmic variants with lower memory consumption. More precisely, it is possible to reduce the memory consumption of the Moulinec–Suquet algorithms by 50%. We underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale. Copyright © 2015 John Wiley & Sons, Ltd.
158 citations
TL;DR: A jump-diffusion model for a single-asset market is considered and results showing the quadratic convergence of the methods are given for Merton's model and Kou's model.
158 citations
TL;DR: The present method performs well in solving the two-dimensional Burgers' equations in fully implicit finite-difference form and is examined by comparison with other analytical and numerical results.
158 citations
TL;DR: In this article, a mathematical model to describe the pyrolysis of a single solid particle of biomass is developed by incorporating improvements in the existing model reported in literature, which couples the heat transfer equation with the chemical kinetics equations.
158 citations