Convergent Difference Schemes for Degenerate Elliptic and Parabolic Equations: Hamilton--Jacobi Equations and Free Boundary Problems
Abstract
Convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented. Simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm. Explicit schemes endowed with an explicit CFL condition are built for time-dependent equations and are used to solve stationary equations iteratively. Explicit and implicit formulations of monotonicity for first- and second-order equations are unified. Bounds on orders of accuracy are established. An example of a scheme which is stable, but nonmonotone and nonconvergent, is presented. Schemes for Hamilton--Jacobi equations, obstacle problems, one-phase free boundary problems, and stochastic games are built and computational results are presented.
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References
1.
D. Aronson, The stability of finite difference approximations to second order linear parabolic differential equations, Duke Math. J., 30 (1963), 117–127
2.
Martino Bardi, Italo Capuzzo‐Dolcetta, Optimal control and viscosity solutions of Hamilton‐Jacobi‐Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., 1997xviii+570, With appendices by Maurizio Falcone and Pierpaolo Soravia
3.
M. Bardi, M. Crandall, L. Evans, H. Soner, P. Souganidis, Viscosity solutions and applications, Lecture Notes in Mathematics, Vol. 1660, Springer‐Verlag, 1997x+259, Lectures given at the 2nd C.I.M.E. Session held in Montecatini Terme, June 12–20, 1995; Edited by I. Capuzzo Dolcetta and P. L. Lions Fondazione C.I.M.E. [C.I.M.E. Foundation]
4.
Guy Barles, Solutions de viscosité des équations de Hamilton‐Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications], Vol. 17, Springer‐Verlag, 1994x+194
5.
G. Barles, P. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271–283
6.
G. Barles, Panagiotis Souganidis, On the large time behavior of solutions of Hamilton‐Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925–939
7.
J. Bramble, B. Hubbard, Vidar Thomée, Convergence estimates for essentially positive type discrete Dirichlet problems, Math. Comp., 23 (1969), 695–709
8.
Luis Caffarelli, Juan Vazquez, Viscosity solutions for the porous medium equation, Proc. Sympos. Pure Math., Vol. 65, Amer. Math. Soc., Providence, RI, 1999, 13–26
9.
R. Courant, K. Friedrichs, H. Lewy, On the partial difference equations of mathematical physics, IBM J. Res. Develop., 11 (1967), 215–234
10.
M. Crandall, P.‐L. Lions, Two approximations of solutions of Hamilton‐Jacobi equations, Math. Comp., 43 (1984), 1–19
11.
Michael Crandall, Hitoshi Ishii, Pierre‐Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1–67
12.
Michael Crandall, Pierre‐Louis Lions, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numer. Math., 75 (1996), 17–41
13.
Michael Crandall, Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1–21
14.
Michael Crandall, Luc Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385–390
15.
Lawrence Evans, Partial differential equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, 1998xviii+662
16.
M. Falcone, R. Ferretti, Semi‐Lagrangian schemes for Hamilton‐Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys., 175 (2002), 559–575
17.
Sigal Gottlieb, Chi‐Wang Shu, Eitan Tadmor, Strong stability‐preserving high‐order time discretization methods, SIAM Rev., 43 (2001), 89–112
18.
A. Harten, J. M. Hyman, and P. D. Lax, Onfinite‐difference approximations and entropy conditions forshocks, Comm. Pure Appl. Math., 29 (1976), pp. 297–322.
19.
Ami Harten, Stanley Osher, Uniformly high‐order accurate nonoscillatory schemes. I, SIAM J. Numer. Anal., 24 (1987), 279–309
20.
R. Jensen, P.‐L. Lions, P. Souganidis, A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. Amer. Math. Soc., 102 (1988), 975–978
21.
Fritz John, On integration of parabolic equations by difference methods. I. Linear and quasi‐linear equations for the infinite interval, Comm. Pure Appl. Math., 5 (1952), 155–211
22.
Inwon Kim, Uniqueness and existence results on the Hele‐Shaw and the Stefan problems, Arch. Ration. Mech. Anal., 168 (2003), 299–328
23.
Hung Kuo, N. Trudinger, On the discrete maximum principle for parabolic difference operators, RAIRO Modél. Math. Anal. Numér., 27 (1993), 719–737
24.
Hung Kuo, Neil Trudinger, Discrete methods for fully nonlinear elliptic equations, SIAM J. Numer. Anal., 29 (1992), 123–135
25.
Hung‐Ju Kuo, Neil Trudinger, Maximum principles for difference operators, Lecture Notes in Pure and Appl. Math., Vol. 177, Dekker, New York, 1996, 209–219
26.
Hung‐Ju Kuo, Neil Trudinger, Positive difference operators on general meshes, Duke Math. J., 83 (1996), 415–433
27.
Chi‐Tien Lin, Eitan Tadmor, High‐resolution nonoscillatory central schemes for Hamilton‐Jacobi equations, SIAM J. Sci. Comput., 21 (2000), 2163–2186
28.
T. Motzkin, W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Physics, 31 (1953), 253–259
29.
Adam Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), 365–379
30.
Adam Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), 1217–1230
31.
Stanley Osher, Chi‐Wang Shu, High‐order essentially nonoscillatory schemes for Hamilton‐Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907–922
32.
J. Sethian, Fast marching methods, SIAM Rev., 41 (1999), 199–235
33.
Panagiotis Souganidis, Approximation schemes for viscosity solutions of Hamilton‐Jacobi equations, J. Differential Equations, 59 (1985), 1–43
34.
Yen‐Hsi Tsai, Li‐Tien Cheng, Stanley Osher, Hong‐Kai Zhao, Fast sweeping algorithms for a class of Hamilton‐Jacobi equations, SIAM J. Numer. Anal., 41 (2003), 673–694
35.
Olof Widlund, Stability of parabolic difference schemes in the maximum norm, Numer. Math., 8 (1966), 186–202
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SIAM Journal on Numerical Analysis
Pages: 879 - 895
ISSN (online): 1095-7170
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Copyright © 2006 Society for Industrial and Applied Mathematics.
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Published online: 25 July 2006
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