Journal of Differential Geometry

Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws

J. A. Sethian

Full-text: Open access

Article information

Source
J. Differential Geom., Volume 31, Number 1 (1990), 131-161.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1214444092

Digital Object Identifier
doi:10.4310/jdg/1214444092

Mathematical Reviews number (MathSciNet)
MR1030668

Zentralblatt MATH identifier
0691.65082

Subjects
Primary: 65P05
Secondary: 65D15: Algorithms for functional approximation

Citation

Sethian, J. A. Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws. J. Differential Geom. 31 (1990), no. 1, 131--161. doi:10.4310/jdg/1214444092. https://projecteuclid.org/euclid.jdg/1214444092


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References

  • [1] K. A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, Princeton, NJ, 1978.
  • [2] A. J. Chorin, Flame advection and propagation algorithms, J. Comput. Phys. 35 (1980) 1.
  • [3] A. J. Chorin, Curvature and solidification, J. Comput. Phys. 58 (1985) 472.
  • [4] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983) 1.
  • [5] B. Engquist and S. Osher, Stable and entropy satisfying approximations for transonic flow calculations, Math. Comp. 34 (1980) 45.
  • [6] B. Engquist and S. Osher & R. Sommerville, Large-scale computations in fluid mechanics, Vols. I and II, Lectures in Applied Math., Amer. Math. Soc, Providence, RI, 1985.
  • [7] C. L. Epstein and M. I. Weinstein, A stable manifold theorem for the curveshortening equation, Comm. Pure Appl. Math. 40 (1987) 119.
  • [8] M. L. Frankel and G. I. Sivashinsky, The effect of viscosity on hydrodynamic stability of a plane flame front, Comb. Sci. Tech. 29 (1982) 207.
  • [9] M. Gage, An isoperimetric inequality with applications to curveshortening, Duke Math. J. 50 (1983) 1225.
  • [10] M. Gage, Curve shortening makes convex curves circular, Invent. Math. 76 (1984) 357.
  • [11] M. Gage and R. S. Hamilton, The equation shrinking convexplanes curves, J. Differential Geometry 23 (1986) 69.
  • [12] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of conservation laws, Comm. Pure Appl. Math. 18 (1965) 697.
  • [13] S. K. Godunov, Finite difference method for numerical computation of discontinuous solutions of the equations of fluid mechanics, Mat. Sb. 47, (1959) 271.
  • [14] M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geometry 26 (1987) 285.
  • [15] M. Grayson, A short note on the evolution of a surfaces via mean curvature, preprint, Department of Mathematics, Stanford University, 1987.
  • [16] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geometry 20 (1984) 237.
  • [17] L. Landau, On the theory of slow combustion, Acta Physiocochimica URSS 19 (1944) 77.
  • [18] J. S. Langer, Instabilities and patternformation in crystal growth, Rev. Modern Phys. 52 (1980) 1.
  • [19] J. S. Langer and H. Muller-Krumhaar, Mode selection in a dendrite-like nonlinear system, Phys. Rev. A 27 (1983) 499.
  • [20] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numericalcomputations, Comm. Pure Appl. Math. 7 (1954) 159.
  • [21] G. H. Markstein, Experimental and theoretical studies of flame front stability, J. Aero. Sci. 18 (1951), 199.
  • [22] G. H. Markstein, Non-steady flame propagation, Pergamon Press and MacMillan, New York, 1964.
  • [23] O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Trudy Moscow Mat. Obsc. 5 (1956) 433.
  • [24] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988) 12-49.
  • [25] B. R. Pamplin, Crystalgrowth, Pergamon Press, New York, 1975.
  • [26] J. Rubinstein, P. Sternberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, preprint, Department of Math., Stanford University, 1987.
  • [27] J. A. Sethian, An analysis of flame propagation, Ph.D. Dissertation, University of California, Berkeley, 1982; CPAM Rep. 79.
  • [28] J. A. Sethian, Curvature and the evolution of fronts, Comm. Math. Phys. 101 (1985) 487.
  • [29] J. A. Sethian, Numerical methods for propagation fronts, in Variational Methods for Free Surface Interfaces (P. Concus and R. Finn, eds.), Springer, New York, 1987.
  • [30] J. A. Sethian and S. Osher, Hamilton-Jacobi based algorithms for interface-media interactions, J. Comput. Phys., to be submitted.
  • [31] G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames. I, Acta Astronautica, 4 (1977) 1177.
  • [32] G. Sod, Numerical methods in fluid dynamics, Cambridge University Press, New York, 1985.
  • [33] D. Turnbull, Phase changes, in Solid State Physics. 3 (F. Sietz and D. Turnbull, eds.), Academic Press, New York, 1956.
  • [34] Y. B. Zeldovich, Structure and stability of steady laminar flame at moderately large Reynolds numbers, Comb. Flame 40 (1981) 225.