## Meshless method for phase field equations

### March 20, 2008

Title: Solving phase field equations using a meshless method

Authors: J X Zhou and M E Li

Abstract:

The phase field equation is solved by using a meshless reproducing kernel particle method (RKPM) for the very first time. The 1D phase field equation is solved using different grid sizes and various time steps at a given grid size. The method can give accurate solutions across the interface, and allows a larger time step than explicit finite-difference method. The 2D phase field equation is computed by the present method and a classic shrinking of a circle is simulated. This shows the powerfulness and the potential of the method to treat more complicated problems.

## DFT studies on magnetoelectric multiferroics

### September 27, 2007

Title: Density functional studies of multiferroic magnetoelectrics

Author
: Nicola A Hill

Abstract:

Multiferroic magnetoelectrics are materials that are both ferromagnetic and ferroelectric in the same phase. As a result, they have a spontaneous magnetization that can be switched by an applied magnetic field and a spontaneous polarization that can be switched by an applied electric field. In this paper we show that density functional theory has been invaluable both in explaining the properties of known magnetically ordered ferroelectric materials, and in predicting the occurrence of new ones. Density functional calculations have shown that, in general, the transition metal d electrons essential for magnetism reduce the tendency for off-center ferroelectric distortion. Consequently, an additional electronic or structural driving force must be present for ferromagnetism and ferroelectricity to occur simultaneously.

## Iterative methods — templates for!

### September 8, 2007

Everybody needs an iterative solver at some point or other. Recently, I sent a few links to a friend of mine. I thought I will also maintain the list in this page for future use.

In case you know of any good resources that I have missed here, leave a note!

## Stability of Crank-Nicholson for variable diffusivity

### September 7, 2007

Title: Stability analysis of the Crank-Nicholson method for variable coefficient diffusion equation

Abstract:
The Crank-Nicholson method is a widely used method to obtain numerical approximations to the diffusion equation due to its accuracy and unconditional stability.

When the diffusion coefficient is not a constant, the general approach is to obtain a discretization for the PDE in the same manner as the case for constant coefficients. In this paper, we show that the manner of this discretization may impact the stability of the resulting method and could lead to instability of the numerical solution. It is shown that the classical Crank-Nicholson method will fail to be unconditionally stable if the diffusion coefficient is computed at the time gridpoints instead of at the midpoints of the temporal subinterval. A numerical example is presented and compared with the exact analytical solution to examine its divergence.

## On discretization of the n-dimensional Laplacian

### June 12, 2007

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

Author(s): Anand Kumar

A discretization of the n-dimensional Laplacian employing all the $3^n$ 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 $\frac{1} {2}$ and 1, respectively.

## Moving mesh spectral method for phase field simulations

### June 12, 2007

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

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

### June 12, 2007

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

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.