#
The Stability Analysis of Crystal Growth

- D. I. Meiron
- D. C. Sorensen
- B. Wedeman
- C. Yang

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Imagine that a piece of crystal is placed in some under-cooled liquid.
As the crystal freezes, the interface between the solid and the liquid
starts to change. The solid phase grows rapidly from the crystal by sending
out branching fingers. This phenomenon is also called * dendritic
solidification *, and is responsible for the complicated interface
observed in snowflakes.

There has been a great deal of interest in the simulation and modeling
of crystal growth and dentridic solidification in the past few years.
It is well known that the change of the temperature of both the crystal
and the liquid can be described by a set of diffusion equations. At the
solid-liquid interface the motion of the interface and the temperature of
these two materials are related by some conservation law.

Both analytical and numerical solutions of the above problem
are difficult to obtain because of the moving boundary. We are interested
in analyzing the stability of a well known stationary solution
that corresponds to a simple parabolic shaped moving front. The
standard techniques of linear stability analysis lead to an eigenvalue
problem. In particular, the rightmost eigenvalues of some convection
diffusion operator are of interest. The stationary solution is said to be
stable if there is no eigenvalue with positive real part.

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## Strategy

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A stationary solution corresponding to a parabolic moving
front can be derived by writing the diffusion equation in a moving frame.
This is known as the * Ivantsov parabola * solution. The standard
linear stability analysis technique yields an eigenvalue problem of
the form
where ** L ** is some convection diffusion operator. The eigenvalue
problem is then solved numerically by discretizing **L** using a
second order finite difference scheme and computing the rightmost eigenvalues
of the corresponding algebraic eigenvalue problem.

## Tools

This problem is challenging because we expect the spectrum to be
unbounded. As the mesh size of the finite difference becomes small,
it is not practical to use the conventional QR type of method to compute
the large scale eigenvalue problem. Since the matrix derived from
finite difference is sparse and structured, fast iterative methods
such as the * Arnoldi * method is attractive in this setting.
The particular software package we used in our computation is
ARPACK , a package that implements the * Implicitly Restarted
Arnoldi Iteration *. We are able to obtain the desired
eigenvalues and eigenvectors of the matrix of order 10,000 or larger in
a reasonable amount of time.

## Results

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Click here to see computed eigenvalues and eigenfunctions.

## Report

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Numerical Computation of Linear Stability of the Crystal Growth
TR96-04, Department of Computational and Applied Mathematics, Rice
University, Houston, TX 77005. Also appears in * Proceedings of the
Copper Mountain Conference on Iterative Methods *, April 9-13, 1996.