Evolutionary partial differential equations for biomedical image processing
Journal of Biomedical Informatics
Evolutionary partial differential equations for biomedical image processing
Computers and Biomedical Research
Geometrical image segmentation by the Allen-Cahn equation
Applied Numerical Mathematics
Flux-based level set method: A finite volume method for evolving interfaces
Applied Numerical Mathematics
A Higher Order Scheme for a Tangentially Stabilized Plane Curve Shortening Flow with a Driving Force
SIAM Journal on Scientific Computing
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The numerical approximation schemes for solving the nonlinear initial value problem $\partial_t b(v) = (A v_x)_x+(Bv)_x+Dv+G$ 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 (1989), 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.