Evolution of Convex Plane Curves Describing Anisotropic Motions of Phase Interfaces
Karol Mikula, Jozef Kačur · SIAM Journal on Scientific Computing · 1996
The numerical approximation schemes for solving the nonlinear initial value problem \[ \partial _t b(v) = \left( {Av_x } \right)_x + \left( {Bv} \right)_x + Dv + G \] with periodic boundary conditions are presented. We assume that the function b is increasing, and asymptotically $b'(s) = 0$ and $b'(s) = + \infty $, so that the model describes in a sense both slow and fast diffusions. The solution also may blow up in a finite time. This problem arises from the evolving curves theory, which was used in the construction of models of motion of phase interface in multiphase thermomechanics by Angenent and Gurtin [Arch. Rational Mech. Anal., 108 (198), pp. 323–391]. The so-called “curve shortening equation” is included in the model. Our approximating solutions converge strongly in $L_2 (I,V)$ space to the weak solution. We also derive an “error estimate”for semidiscretization, which implies uniqueness. The numerical experiments in various situations of the “anisotropic curve shortening” are discussed.