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Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. (English) Zbl 0659.65132

New methodologies are formulated for following fronts propagating with curvature-dependent speed, which models, among other things, crystal growth and flame propagation. The algoritms are based on an approximation of the Hamilton-Jacobi formulations for such problems. The paper is carefully written with numerical exemplification (including detailed graphics) which shows that the numerical schemes accurately capture the formation of sharp gradients and cusps in the moving fronts.
Reviewer: R.S.Anderssen

MSC:

65Z05 Applications to the sciences
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35L65 Hyperbolic conservation laws
80A25 Combustion
82D25 Statistical mechanics of crystals
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[1] Barles, G., (Report No. 464 (1985), Institut National de Recherche en Informatique et en Automatique (INRIA): Institut National de Recherche en Informatique et en Automatique (INRIA) Sophia Antipolis, France), (unpublished)
[2] Brakke, K. A., The Motion of a Surface by Its Mean Curvature (1978), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0386.53047
[3] Chorin, A. J., J. Comput. Phys., 57, 472 (1985)
[4] Chorin, A. J., J. Comput. Phys., 35, 1 (1980)
[5] Crandall, M. G.; Lions, P. L., Math. Comp., 43, 1 (1984)
[6] Frankel, M. L.; Sivashinsky, G. I., Comb. Sci. Technol., 29, 207 (1982)
[7] Gage, M., Duke Math. J., 50, 1225 (1983)
[8] Gage, M., Invent. Math., 76, 357 (1984)
[9] Gage, M.; Hamilton, R. S., J. Differential Geom., 23, 69 (1986)
[10] Grayson, M., J. Differential Geom., 26, 285 (1988)
[11] Grayson, M., A Short Note on the Evolution of a Surfaces via Mean Curvature (1987), Stanford University Mathematics Dept, preprint
[12] Godunov, S. K., Mat. Sb., 47, 271 (1959)
[13] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., J. Comput. Phys., 71, 231 (1987)
[14] Harten, A.; Osher, S.; Engquist, B.; Chakravarthy, S., Appl. Num. Math., 2, 237 (1986)
[15] Huisken, G., J. Differential Geom., 20, 237 (1984)
[16] Kutnetsov, N. N., (Miller, J. J.H., Topics in Numerical Analysis III (1977), Academic Press: Academic Press New York)
[17] Kruz’kov, S. N., Math. U.S.S.R. Sb., 10, 217 (1970)
[18] Landau, L., Acta Physiocochim. URSS, 19 (1944)
[19] Langer, J. S., Rev. Mod. Phys., 52, 1 (1980)
[20] Langer, J. S.; Muller-Krumhaar, H., Phys. Rev. A, 27, 499 (1983)
[21] Lay, P. D., Comm. Pure Appl Math., 10, 537 (1957)
[22] Markstein, G. H., J. Aero. Sci., 18, 199 (1951)
[23] Markstein, G. H., Non-Steady Flame Propagation (1964), Pergammon: Pergammon MacMillan C., New York · Zbl 0077.19201
[24] Mullins, W. W.; Sekerka, R. F., J. Appl Phys., 34, 2885 (1963)
[25] Nichols, F. A.; Mullins, W. W., Trans. Metall Soc. RIME, 223, 1840 (1965)
[26] Noh, W.; Woodward, P., (van de Vooran, A. I.; Zandberger, P. J., Proceedings, Fifth International Conference on Fluid Dynamics (1976), Springer-Verlag: Springer-Verlag New York/Berlin)
[27] Osher, S., SIAM J. Num. Anal., 21, 217 (1984)
[28] Oleinik, O. A., T. Moscow Mat. Obsc., 5, 433 (1956)
[29] Situ, C.; Osher, S., J. Comput. Phys., 77, 439 (1988)
[30] Pamplin, B. R., Crystal Growth (1975), Pergammon: Pergammon New York
[31] Sethian, J. A., (CPAM Rep. 79. CPAM Rep. 79, Ph.D. dissertation (June 1982), University of California: University of California Berkeley, California)
[32] Sethian, J. A., Commun. Math. Phys., 101, 487 (1985)
[33] Sethian, J. A., (Concus, P.; Finn, R., Variational Methods for Free Surface Interfaces (1987), Springer-Vertag: Springer-Vertag New York)
[34] Sethian, J. A., J. Comput. Phys., 54, 425 (1984)
[35] Sethian, J. A., (Engquist, B.; Majda, A.; Osher, S., Computational Fluid Mechanics and Reacting Gas Flows (1986), Institute for Mathematics and Its Applicationsm Univ. of Minnesota)
[36] J.A. Sethian and S. Osher, Level Set Algorithms for Hele-Shaw Flow, J. Comput. Phys., in preparation.; J.A. Sethian and S. Osher, Level Set Algorithms for Hele-Shaw Flow, J. Comput. Phys., in preparation.
[37] Sivashinsky, G. I., Acta Astronaut., 4, 1177 (1977)
[38] Turnbull, D., (Seitz, F.; Turnbull, D., Solid State Physics I, 3 (1956), Academic Press: Academic Press New York)
[39] Zabusky, N. J.; Overman, E. A., J. Comput. Phys., 52, 351 (1984)
[40] Zeldovich, Y. B., Comb. Flame, 40, 225 (1981)
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