In this section, the reason of applying CCD scheme to predict the motion of curves will be explained through several studies in this subsection. Initially, the shape of the curve is given as
p(x x0)2+ (y y0)2 = 2.0 + 1.0 cos(!1✓) + 0.5 sin(!2✓) (35)
where ✓ = tan 1[(y y0)/(x x0)]. The degree of complexity of the curve is determined
Figure 12: Task 2. Comparison of the solutions sought at di↵erent times by applying the CCD scheme and the second-order centered di↵erence scheme. The computations are carried out with h = dy = dx = 0.05 and dt = 0.01⇥ h. (a) t = 0.0625; (b) t = 0.250
by the coefficients, !1 and !2. The larger values of !1 and !2, the larger change of curvatures along the curve. In a box [10,10], the center of the curve is located at (x0, y0) = (5, 5).
6.4.1 Task 1, (!1, !2) = (0, 0)
Investigation of this case is to show that either CCD or second-order centered di↵erence scheme is applicable to disctetize the spatial derivatives. Two solutions remain smooth and are identical with each other. In this task, the choice of !1 = !2 = 0 results in a circle. As it is shown in figure 11, two solutions are identical during the evolution of two curves. Constant curvature is seen along the curve. Both CCD and second-order centered schemes yield accurately simulated evolutions.
6.4.2 Task 2, (!1, !2) = (5, 6)
Another task involves variable curvatures by setting !1 = 5 and !2 = 6 in (35). Accord-ing to the snap shot of the predicted curve, the solution sought from the second-order centered scheme is more dissipated in the vicinity of the convex and concave parts of the curve. In other words, application of the second-order centered di↵erence scheme yields a smoother solution due to a largely introduced discretization error in the ap-proximation of spatial derivatives. By applying the CCD scheme, one can calculate the curvature term more accurately than that using the second-order centered scheme. As a result, the solution obtained from the CCD scheme is less polluted by the dissipation
(a) (b)
Figure 13: Task 3. Comparison of the solutions sought at di↵erent times by applying the CCD scheme and the second-order centered di↵erence scheme. The computations are carried out with h = dy = dx = 0.05 and dt = 0.01⇥ h. (a) t = 0.0625; (b) t = 0.250
error introduced to regions with a large curvature, or along the convex and concave parts of the curve.
6.4.3 Task 3, (!1, !2) = (5, 12)
Figure 13 shows the predicted motion of the curve with a more complicated curvature than the curve investigated in task 2. As we mentioned earlier, the solutions obtained from the second-order centered di↵erence scheme are smeared more quickly than CCD scheme does in the sense of introducing more artificial viscosity in the coarse of the simulation of Eq.(9).
In summary, it is advantageous to apply the CCD scheme to capture more accurately the motion of curves with less dissipation error being added to the convex and concave parts of curve. Second-order centered di↵erence scheme also provides a good prediction of the motion of curve in case that the curvature does not change rapidly along the curve. If the shape of curve is too complicated, second-order centered di↵erence scheme is not recommended especially in the very beginning phase of the calculation.
7 Concluding remarks
Within the framework of semi-discretization schemes, an optimal third-order accurate TVD Runge-Kutta temporal scheme for time derivative term and a fifth-order accurate
have been developed in a three-point grid stencil. This computationally efficient finite di↵erence scheme has been applied to simulate mean curvature driven motion of curves, subject to di↵erent initial curves, with great success. Our main objective of this study is to enlighten how two di↵erent mechanisms control the evolution of curve and how they are related to the curvature of the curve under investigation. In the evolution, the surface tension driven curve is subject all the time to the damping e↵ect applied on each point of the curve.
References
[1] S. Hildebrandt, H. Karcher (Eds.). Geometric Analytic and Nonlinear Partial Dif-ferential Equations. Springer-Verlag Berlin Heidelberg. New York; 2003; ISBN 3-540-44051-8.
[2] U. Dierkes, S. Hildebrandt, F. Sauvigny. Minimal Surface, Grundlehren der Math-ematischen Wissenschaften. Berlin Springer-Verlag; 1992.
[3] P. Smereka. Semi-implicit level set methods for curvature and surface di↵usion motion. Journal of Scientific Computing; 2003; 19:439-456.
[4] S. Osher, R. P. Fedkiw. Level set methods: An overview and some recent results.
Journal of Computational Physics; 2001; 169:463-502.
[5] J.A. Sethian. Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechan-ics, Computer Vision, and Materials Science. Cambridge University Press; 1996.
[6] J.A. Sethian, P. Smereka. Level set methods for fluid interface. Ann. Rev. Fluid Mech.; 2003; 35:341-372.
[7] D. L. Chopp. Motion by intrinsic Laplacian of curvature. Interfaces and Free Boundary; 1999; 1:107-123.
[8] M. Khenner, A. Averbuch, M. Israeli, M. Nathan. Numerical simulation of grain-boundary grooving by level set method. Journal of Computational Physics; 2001;
170:764-784.
[9] T. Tasdizen, R. Whitaker, P. Burchard, S. Osher. Geometric Surface Processing via Normal Maps. UCLA CAM Report; 2002; 02-03.
[10] C. W. Shu, S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics; 1968; 77:439-461.
[11] S. Gottlieb, C.W. Shu. Total variation diminishing Runge-Kutta schemes. Mathe-matics of Computation; 1998; 67:73-85.
[12] P. C. Chu., C. Fan. A three point combined compact di↵erence scheme. Journal of Computational Physics; 1998; 140: 370-399.
[13] M. Sussman, P. Smereka, S. Osher. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics; 1994; 114:146-159.
[14] X. D. Liu, S. Osher, T. Chan. Weighted essential non-oscillatory schemes. Journal of Computational Physics; 1994; 115:200-212.
[15] G. S. Jiang, C.W. Shu. Efficient implementation of weighted ENO schemes. Journal of Computational Physics; 1996; 126:202-228.
[16] L. T. Cheng, R. Tsai. Redistancing by flow of time dependent eikonal equation.
Journal of Computational Physics; 2008; 227:4002-4017.
[17] C.W. Shu. High order ENO and WENO schemes for computational fluid dynam-ics. High-Order Methods for Computational Physdynam-ics. T.J.Barth and H. Deconinck, editors. Lecture Notes in Computational Science and Engineering. Springer; 1999;
9:439-582.
[18] S. Osher, J. A. Sethian. Fronts propagating with curvature-dependent speed: al-gorithms based on Hamilton-Jacobian formulations. Journal of Computational Physics; 1988; 79:12-49.
[19] D. Hartmann, M. Meinke, W. Schroder, Di↵erential equation based constrained reinitialization for level set methods, Journal of Computational Physics; 2008;
227:6821-6845.
[20] G. Russo, P. Smereka, A remark on computing distance functions, Journal of Com-putational Physics; 2000; 163:51-67.