Computational analysis of the conserved curvature driven flow for open curves in the plane

The paper studies the constrained curvature flow for open planar curves with fixed endpoints by means of its numerical solution. This law originates in the theory of phase transitions for crystalline materials and where it describes the evolution of closed embedded curves with constant enclosed area. We show that the area is preserved for open curves with fixed endpoints as well. Here, the area is given by the curve and its ends connected to the origin of coordinates. We provide the form of the stationary solution towards which any other solution converges asymptotically in time. The evolution law is reformulated by means of the direct method into the system of degenerate parabolic partial differential equations for the curve parametrization. This system is spatially discretized by means of the flowing finite volumes method and solved numerically by the explicit Runge–Kutta solver. We experimentally investigate the order of approximation of the scheme by means of our numerical data and by knowing the analytical solution. We also discuss the role of the suitable tangential redistribution. For this purpose, several computational studies related to the open curve dynamics are presented.