On discretization of the n-dimensional Laplacian

Title: A discretization of the n-dimensional Laplacian for a dimension-independent stability limit

Author(s): Anand Kumar

Source: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 457, No. 2015, (Nov. 8, 2001), pp. 2667-2674.
Abstract:

A discretization of the n-dimensional Laplacian employing all the points of the finite-difference stencil has been presented. Using this discretization, the stability limit of the heat conduction and the wave equations are found to become dimension-independent and 1, respectively.

Moving mesh spectral method for phase field simulations

Title: Spectral implementation of an adaptive moving mesh method for phase-field equations

Authors: W M Feng, P Yu, S Y Hu, Z K Liu, Q Du and L-Q Chen

Source: Journal of Computational Physics, Vo. 220, Issue 1, 20 December 2006, pp. 498-510.

Abstract:

Phase-field simulations have been extensively applied to modeling microstructure evolution during various materials processes. However, large-scale simulations of three-dimensional (3D) microstructures are still computationally expensive. Among recent efforts to develop advanced numerical algorithms, the semi-implicit Fourier spectral method is found to be particularly efficient for systems involving long-range interactions as it is able to utilize the fast Fourier transforms (FFT) on uniform grids. In this paper, we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy, while maintaining a uniform grid in the computational domain for the spectral implementation. This approach not only provides more accurate treatment at the interfaces requiring higher resolution, but also retains the numerical efficiency of the semi-implicit Fourier spectral method. Numerical examples using the new adaptive moving mesh semi-implicit Fourier spectral method are presented for both two and three space dimensional microstructure simulations, and they are compared with those obtained by other methods. By maintaining a similar accuracy, the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size.

Finite difference schemes for Cahn-Hilliard equations

Title: Numerical study of the Cahn-Hilliard equation in one, two and three dimensions

Authors: E V L de Mello and Otton Teixeira da Silveira Filho

Source: Physica A: Statistical and theoretical physics, Vol. 347, 1 March 2005, pp. 429-443.

Abstract:

The Cahn–Hilliard (CH) equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff and the difficulty to solve it numerically increases with the dimensionality and therefore, there are several published numerical studies in one dimension (1D), dealing with different approaches, and much fewer in two dimensions (2D). In three dimensions (3D) there are very few publications, usually concentrate in some specific result without the details of the used numerical scheme. We present here a stable and fast conservative finite difference scheme to solve the CH with two improvements: a splitting potential into an implicit and explicit in time part and the use of free boundary conditions. We show that gradient stability is achieved in one, two and three dimensions with large time marching steps than normal methods.