Positivity-preserving numerical schemes for lubrication-type equations

SIAM Journal on Numerical Analysis

Volumn 37, Issue 2, 2000, Pages 523-555

  Zhornitskaya, L ^a   Bertozzi, A L ^a  

^a DUKE UNIVERSITY   (United States)

Author keywords

Entropy dissipation; Finite differences; Finite elements; Fourth order degenerate diffusion; Lubrication equations; Positivity preserving

Indexed keywords

EID: 0000708355     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0036142998335698     Document Type: Article

References (23)

1. H. P. GREENSPAN, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84 (1978), pp. 125-143.
2. P. CONSTANTIN, T. F. DUPONT, R. E. GOLDSTEIN, L. P. KADANOFF, M. J. SHELLEY, AND S.-M. ZHOU, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E, 47 (1993), pp. 4169-4181.
3. C. M. ELLIOTT AND H. GARCKE, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), pp. 404-423.
4. M. A. LEWIS, Spatial coupling of plant and herbivore dynamics: The contribution of herbivore dispersal to transient and persistent "waves" of damage, Theoret. Population Biol., 45 (1994), p. 277.
5. G. GRÜN, Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening, Z. Anal. Anwendungen, 14 (1995), pp. 541-574.
6. F. BERNIS, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations, 1 (1996), pp. 337-368.
7. F. BERNIS, Finite speed of propagation for thin viscous flows when 2 ≤ n < 3, C. R. Acad. Sci. Paris, 322 (1996), pp. 1169-1174.
8. F. BERNIS, Change in sign of the solutions to some parabolic problems, in Nonlinear Analysis and Applications (Arlington, TX, 1986), Lecture Notes in Pure and Appl. Math. 109, V. Lakshmikantham, ed., Marcel Dekker, New York, 1987, pp. 75-82.
9. F. BERNIS AND A. FRIEDMAN, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), pp. 179-206.
10. A. L. BERTOZZI, M. P. BRENNER, T. F. DUPONT, AND L. P. KADANOFF, Singularities and similarities in interface flow, in Trends and Perspectives in Applied Mathematics, L. Sirovich, ed., Appl. Math. Sci. 100, Springer-Verlag, New York, 1994, pp. 155-208.
11. A. L. BERTOZZI, Symmetric singularity formation in lubrication-type equations for interface motion, SIAM J. Appl. Math., 56 (1996), pp. 681-714.
12. R. ALMGREN, A. L. BERTOZZI, AND M. P. BRENNER, Stable and unstable singularities in the unforced Hele-Shaw cell, Phys. Fluids, 8 (1996), pp. 1356-1370.
13. A. L. BERTOZZI AND M. PUGH, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), pp. 85-123.
14. E. BERETTA, M. BERSTCH, AND R. DAL PASSO, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), pp. 175-200.
15. A. L. BERTOZZI AND M. P. BRENNER, Linear stability and transient growth in driven contact lines, Phys. Fluids, 9 (1997), pp. 530-539.
16. A. L. BERTOZZI, Loss and gain of regularity in a lubrication equation for thin viscous films, in Free Boundary Problems: Theory and Applications, Proceedings of the International Colloquium on Free Boundary Problems, Toledo, Spain, June 1993, J. I. Díaz, M. A. Herrero, A. Liñán, and J. L. Vázquez, eds., Pitman Res. Notes Math. Ser. 323, Longman House, London, 1995, pp. 72-85.
17. J. W. BARRETT, J. F. BLOWEY, AND H. GARCKE, Finite element approximation of a fourth order degenerate parabolic equation, Numer. Math., 77 (1997), pp. 1-34.
18. A. L. BERTOZZI AND M. PUGH, The lubrication approximation for thin viscous films: The moving contact line with a "porous media" cut off of Van der Waals interactions, Nonlinearity, 7 (1994), pp. 1535-1564.
19. M. W. HIRSCH AND S. SMALE, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, San Diego, 1974.
20. A. L. BERTOZZI AND M. C. PUGH, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), pp. 625-661.
21. P. EHRHARD, The spreading of hanging drops, J. Colloid Interface Sci., 168 (1994), pp. 242-246.
22. R. TEMAM, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, New York, 1979.
23. G. GRÜN AND M. RUMPF, Nonnegativity Preserving Numerical Schemes for the Thin Film Equation, preprint, 1998.