Numerical schemes for conservation laws via Hamilton-Jacobi equations

Corrias, L.; Falcone, M.; Natalini, R.

doi:10.1090/S0025-5718-1995-1265013-5

Numerical schemes for conservation laws via Hamilton-Jacobi equations
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by L. Corrias, M. Falcone and R. Natalini PDF

Math. Comp. 64 (1995), 555-580 Request permission

Abstract:

We present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux. The numerical methods described here take their origin from approximation schemes for Hamilton-Jacobi-Bellman equations related to optimal control problems and exhibit several interesting features: the convergence result still holds for quite arbitrary time steps, the main assumption for convergence can be interpreted as a discrete analogue of Oleinik’s entropy condition, numerical diffusion around the shocks is very limited. Some tests are included in order to compare the performances of these methods with other classical methods (Godunov, TVD).

References

M. Bardi and L. C. Evans, On Hopf’s formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal. 8 (1984), no. 11, 1373–1381. MR 764917, DOI 10.1016/0362-546X(84)90020-8
M. Bardi, M. Falcone, and P. Soravia, Fully discrete schemes for the value function of pursuit-evasion games, Advances in dynamic games and applications (Geneva, 1992) Ann. Internat. Soc. Dynam. Games, vol. 1, Birkhäuser Boston, Boston, MA, 1994, pp. 89–105. MR 1273550
Yann Brenier, Un algorithme rapide pour le calcul de transformées de Legendre-Fenchel discrètes, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 20, 587–589 (French, with English summary). MR 1001813
V. Caselles, Scalar conservation laws and Hamilton-Jacobi equations in one-space variable, Nonlinear Anal. 18 (1992), no. 5, 461–469. MR 1152721, DOI 10.1016/0362-546X(92)90013-5

Transformées de Legendre-Frenchel discrètes et applications

Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), no. 2, 487–502. MR 732102, DOI 10.1090/S0002-9947-1984-0732102-X
Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43 (1984), no. 167, 1–19. MR 744921, DOI 10.1090/S0025-5718-1984-0744921-8
M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim. 15 (1987), no. 1, 1–13. MR 866164, DOI 10.1007/BF01442644
Maurizio Falcone and Tiziana Giorgi, An approximation scheme for evolutive Hamilton-Jacobi equations, Stochastic analysis, control, optimization and applications, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, 1999, pp. 289–303. MR 1702966
Ami Harten, On a large time-step high resolution scheme, Math. Comp. 46 (1986), no. 174, 379–399. MR 829615, DOI 10.1090/S0025-5718-1986-0829615-4
Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357–393. MR 701178, DOI 10.1016/0021-9991(83)90136-5
Eberhard Hopf, Generalized solutions of non-linear equations of first order, J. Math. Mech. 14 (1965), 951–973. MR 0182790

First order quasilinear equations in independent variables

81

10

S. N. Kružkov, The Cauchy problem in the large for certain non-linear first order differential equations, Soviet Math. Dokl. 1 (1960), 474–477. MR 0121575

The method of finite differences for a first-order non-linear equation with many independent variables

5

The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the first order with several variables

155

5

Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
Randall J. LeVeque, A large time step generalization of Godunov’s method for systems of conservation laws, SIAM J. Numer. Anal. 22 (1985), no. 6, 1051–1073. MR 811183, DOI 10.1137/0722063
Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
Pierre-Louis Lions and Panagiotis Souganidis, Convergence of MUSCL type methods for scalar conservation laws, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 5, 259–264 (English, with French summary). MR 1071622
O. A. Oleĭnik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 165–170 (Russian). MR 0117408
Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217–235. MR 736327, DOI 10.1137/0721016
Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. MR 965860, DOI 10.1016/0021-9991(88)90002-2