Surface diffusion is a motion of crystals that the normal velocity of an interface
is proportional to the surface Laplacian of mean curvature with respect to the
arclength. It is a very popular issue for materials science and applied mathematics.
Overview this thesis, our numerical method is based on the finite difference
method, compare the numerical results with the results of E. B‥ansch, P. Morin
and R. H. Nochetto, which uses the finite element method for surface diffusion,
and give the proofs of two known properties of surface diffusion, the preservation
of area and the decrease of arclength. Moreover, we also give the sufficient condition
for choosing t in our numerical scheme, which makes sure that if t is
chosen under the condition, the existence and uniqueness of numerical solution is
assured.

[1] E. B‥ansch, P. Morin and R. H. Nochetto, A finite element method for surface
diffusion: the parametric case. Journal of Computational Physics 203 (2005),
pp.321-343.
[2] Xinfu Chen and Jong-Shenq Guo (2010).Motion by curvature of planar curves
with end points moving freely on a line.
[3] C.M Elliott and S. Maier-Paape, Losing a graph with surface diffusion. Hok-
kaido Math. J. 30 (2001) 297-305.
[4] S. H. Friedberg, A. J. Insel and L. E. Spence (2003). LINEAR ALGEBRA (4th
ed.). New Jersey: Prentice Hall, pp. 295-296.
[5] Y. Giga and K. Ito. On pinching of curves moved by surface diffusion. Com-
mun. Appl, Anal. 2 (1998) 393-405.
[6] F. J. Humphreys and M. Hatherly (1995). Recrystallization and related annealing
phenomena, Elsevier.
[7] M.-C. Lai, C.-W. Hsu and H. Huang. A front-tracking method for motion
by mean curvature with surfactant. Adv. Appl. Math. Mech., 2 (2009), pp.
288-300.
[8] W. W. Mullins, Theory of thermal grooving. Journal of Applied Physics,
28(1957), 333-339.
[9] Zhenguo Pan and Brian Wetton. Numerical methods for coupled surface and
grain boundary motion. European Journal of Applied Mathematics (2008),
19: 311-327.