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: In this article, the authors proposed a semi-discrete scheme for phase field dendritic crystal growth, which is derived from the variation of a free energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient-dependent anisotropic coefficient.
Abstract: Summary
We present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model, which is derived from the variation of a free-energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient-dependent anisotropic coefficient. We introduce a novel Invariant Energy Quadratization approach to transform the free-energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations. Based on the reformulated equivalent governing system, we develop a first and a second order semi-discretized scheme in time for the system, in which all nonlinear terms are treated semi-explicitly. The resulting semi-discretized equations consist of a linear elliptic equation system at each time step, where the coefficient matrix operator is positive definite and thus, the semi-discrete system can be solved efficiently. We further prove that the proposed schemes are unconditionally energy stable. Convergence test together with 2D and 3D numerical simulations for dendritic crystal growth are presented after the semi-discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes. Copyright © 2016 John Wiley & Sons, Ltd.
164 citations
TL;DR: In this article, the authors classified one-dimensional, transient, two-phase flow with separate continuity, momentum, and energy equations for each phase by use of the method of characteristics.
Abstract: Equation systems describing one-dimensional, transient, two-phase flow with separate continuity, momentum, and energy equations for each phase are classified by use of the method of characteristics...
164 citations
TL;DR: High-order finite difference methods for solving the Helmholtz equation are developed and analyzed, in one and two dimensions on uniform grids, and a symmetric high-order representation is developed for a Neumann boundary condition.
163 citations
TL;DR: In this article, a numerical algorithm based on the volume of fluid (VOF) technique is used to study the non-linear behavior and damping characteristics of liquid sloshing in a moving partially filled rectangular tank.
163 citations
TL;DR: In this article, a finite difference procedure that reflects the dominance of convection in incompressible flow in porous media is developed. But this method is not suitable for the case of two-phase, incompressibly flow.
Abstract: Two-phase, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Convection physically dominates diffusion, and the object of this paper is to develop a finite difference procedure that reflects this dominance. The pressure equation, which is elliptic in appearance, is discretized by a standard five-point difference method. The concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms. A convergence analysis is given for the method.
163 citations