Figure 1.
Graphs of $ w_k $ , $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w}_k $ and $ \bar{w}_k $ (thick curves)
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On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces
March
2018, 38(3): 1103-1125.
doi: 10.3934/dcds.2018046
Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces
| 1. | Graduate School of Maritime Sciences, Kobe University, Higashinada, Kobe 658-0022, Japan |
| 2. | Yasuna Machine Designing, Hojo-Umehara, Himeji 670-0945, Japan |
We consider a threshold-type algorithm for curvature-dependent motions (CDM for short) of hypersurfaces. This algorithm was numerically studied by Kimura - Notsu [
Mathematics Subject Classification:
Primary: 35K15, 35K55; Secondary: 65M12.
Citation:
Katsuyuki Ishii, Takahiro Izumi. Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces. Discrete & Continuous Dynamical Systems - A,
2018, 38
(3)
: 1103-1125.
doi: 10.3934/dcds.2018046
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References:
| [1] |
G. Barles and C. Georgelin,
A simple proof of convergence for an approximation scheme for computing motion by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500.
doi: 10.1137/0732020. |
| [2] |
J. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, in "Computational Crystal Growers Workshop", J. Taylor ed., Selected Lectures in Math., Amer. Math. Soc., Province, 1992. |
| [3] |
A. Chambolle,
An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195-218.
|
| [4] |
A. Chambolle and M. Novaga,
Approximation of the anisotropic mean curvature flow, Math. Models Methods Appl. Sci., 17 (2007), 833-844.
doi: 10.1142/S0218202507002121. |
| [5] |
A. Chambolle and M. Novaga,
Implicit time discretization of the mean curvature flow with a discontinuous forcing term, Interfaces Free Bound., 10 (2008), 283-300.
|
| [6] |
M.G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. A. M. S., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
| [7] |
S. Esedoḡlu, S.J. Ruuth and R. Tsai,
Diffusion generated motion using the signed distance function, J. Comp. Phys., 229 (2010), 1017-1042.
doi: 10.1016/j.jcp.2009.10.002. |
| [8] |
T. Eto, Y. Giga and K. Ishii,
An area minimizing scheme for anisotropic mean curvature flow, Adv. Differential Equations, 17 (2012), 1031-1084.
|
| [9] |
L.C. Evans,
Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557.
doi: 10.1512/iumj.1993.42.42024. |
| [10] |
L.C. Evans and J. Spruck,
Motion of level sets by mean curvature Ⅱ, Trans. Amer. Math. Soc., 330 (1992), 321-332.
doi: 10.1090/S0002-9947-1992-1068927-8. |
| [11] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. |
| [12] |
K. Ishii and M. Kimura,
Convergence of a threshold-type algorithm using the signed distance function, Interfaces Free Bound., 18 (2016), 479-522.
doi: 10.4171/IFB/371. |
| [13] |
M. Kimura and H. Notsu,
A level set method using the signed distance function, Japan J. Indust. Appl. Math., 19 (2002), 415-446.
doi: 10.1007/BF03167487. |
| [14] |
S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs, 13. Mathematical Society of Japan, Tokyo, 2004. |
| [15] |
F. Leoni,
Convergence of an approximation scheme for curvature-dependent motion of sets, SIAM J. Numer. Anal., 39 (2001), 1115-1131.
doi: 10.1137/S0036142900370459. |
| [16] |
R.Z. Mohammad and K. Švadlenka,
Multiphase volume-preserving interface motion via localized signed distance vector scheme, Discrete and Continuous Dynamical Systems, Series S, 8 (2015), 969-988.
doi: 10.3934/dcdss.2015.8.969. |
| [17] |
L. Vivier,
Convergence of an approximation scheme for computing motions with curvature dependent velocities, Differential Integral Equations, 13 (2000), 1263-1288.
|
show all references
References:
| [1] |
G. Barles and C. Georgelin,
A simple proof of convergence for an approximation scheme for computing motion by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500.
doi: 10.1137/0732020. |
| [2] |
J. Bence, B. Merriman and S. Osher, Diffusion generated motion by mean curvature, in "Computational Crystal Growers Workshop", J. Taylor ed., Selected Lectures in Math., Amer. Math. Soc., Province, 1992. |
| [3] |
A. Chambolle,
An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195-218.
|
| [4] |
A. Chambolle and M. Novaga,
Approximation of the anisotropic mean curvature flow, Math. Models Methods Appl. Sci., 17 (2007), 833-844.
doi: 10.1142/S0218202507002121. |
| [5] |
A. Chambolle and M. Novaga,
Implicit time discretization of the mean curvature flow with a discontinuous forcing term, Interfaces Free Bound., 10 (2008), 283-300.
|
| [6] |
M.G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. A. M. S., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
| [7] |
S. Esedoḡlu, S.J. Ruuth and R. Tsai,
Diffusion generated motion using the signed distance function, J. Comp. Phys., 229 (2010), 1017-1042.
doi: 10.1016/j.jcp.2009.10.002. |
| [8] |
T. Eto, Y. Giga and K. Ishii,
An area minimizing scheme for anisotropic mean curvature flow, Adv. Differential Equations, 17 (2012), 1031-1084.
|
| [9] |
L.C. Evans,
Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557.
doi: 10.1512/iumj.1993.42.42024. |
| [10] |
L.C. Evans and J. Spruck,
Motion of level sets by mean curvature Ⅱ, Trans. Amer. Math. Soc., 330 (1992), 321-332.
doi: 10.1090/S0002-9947-1992-1068927-8. |
| [11] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. |
| [12] |
K. Ishii and M. Kimura,
Convergence of a threshold-type algorithm using the signed distance function, Interfaces Free Bound., 18 (2016), 479-522.
doi: 10.4171/IFB/371. |
| [13] |
M. Kimura and H. Notsu,
A level set method using the signed distance function, Japan J. Indust. Appl. Math., 19 (2002), 415-446.
doi: 10.1007/BF03167487. |
| [14] |
S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs, 13. Mathematical Society of Japan, Tokyo, 2004. |
| [15] |
F. Leoni,
Convergence of an approximation scheme for curvature-dependent motion of sets, SIAM J. Numer. Anal., 39 (2001), 1115-1131.
doi: 10.1137/S0036142900370459. |
| [16] |
R.Z. Mohammad and K. Švadlenka,
Multiphase volume-preserving interface motion via localized signed distance vector scheme, Discrete and Continuous Dynamical Systems, Series S, 8 (2015), 969-988.
doi: 10.3934/dcdss.2015.8.969. |
| [17] |
L. Vivier,
Convergence of an approximation scheme for computing motions with curvature dependent velocities, Differential Integral Equations, 13 (2000), 1263-1288.
|
Figure 2.
Graphs of of $ w^k $ , $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w}^k $ and $ \bar{w}^k $ (thick curves)
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