A numerical method for fractal conservation laws

Droniou, Jérôme

doi:10.1090/S0025-5718-09-02293-5

A numerical method for fractal conservation laws
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by Jérôme Droniou;

Math. Comp. 79 (2010), 95-124

DOI: https://doi.org/10.1090/S0025-5718-09-02293-5

Published electronically: July 29, 2009

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Abstract:

We consider a fractal scalar conservation law, that is to say, a conservation law modified by a fractional power of the Laplace operator, and we propose a numerical method to approximate its solutions. We make a theoretical study of the method, proving in the case of an initial data belonging to $L^\infty \cap BV$ that the approximate solutions converge in $L^\infty$ weak-$*$ and in $L^p$ strong for $p<\infty$, and we give numerical results showing the efficiency of the scheme and illustrating qualitative properties of the solution to the fractal conservation law.

References

Nathaël Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ. 7 (2007), no. 1, 145–175. MR 2305729, DOI 10.1007/s00028-006-0253-z
Alibaud N., Azerad P. and Isebe D., A non-monotone nonlocal conservation law for dune morphodynamics, submitted for publication.
Nathaël Alibaud, Jérôme Droniou, and Julien Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 479–499. MR 2339805, DOI 10.1142/S0219891607001227
Jean-Pierre Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris 256 (1963), 5042–5044 (French). MR 152860
Olivier Alvarez, Philippe Hoch, Yann Le Bouar, and Régis Monneau, Dislocation dynamics: short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal. 181 (2006), no. 3, 449–504. MR 2231781, DOI 10.1007/s00205-006-0418-5
Piotr Biler, Tadahisa Funaki, and Wojbor A. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998), no. 1, 9–46. MR 1637513, DOI 10.1006/jdeq.1998.3458
Piotr Biler, Grzegorz Karch, and Wojbor A. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators, Studia Math. 148 (2001), no. 2, 171–192. MR 1881259, DOI 10.4064/sm148-2-5
Piotr Biler, Grzegorz Karch, and Wojbor A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré C Anal. Non Linéaire 18 (2001), no. 5, 613–637. MR 1849690, DOI 10.1016/S0294-1449(01)00080-4
Jean-Michel Bony, Philippe Courrège, and Pierre Priouret, Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 2, 369–521 (1969) (French). MR 245085
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1016/S1570-8659(00)07005-8
Claire Chainais-Hillairet and Jérôme Droniou, Convergence analysis of a mixed finite volume scheme for an elliptic-parabolic system modeling miscible fluid flows in porous media, SIAM J. Numer. Anal. 45 (2007), no. 5, 2228–2258. MR 2346377, DOI 10.1137/060657236
Raymond H. Chan and Michael K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38 (1996), no. 3, 427–482. MR 1409592, DOI 10.1137/S0036144594276474
Clavin P., Instabilities and nonlinear patterns of overdriven detonations in gases, H. Berestycki and Y. Pomeau (eds.), Nonlinear PDE’s in Condensed Matter and Reactive Flows, Kluwer (2002), 49-97.
J. Droniou, T. Gallouet, and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ. 3 (2003), no. 3, 499–521. Dedicated to Philippe Bénilan. MR 2019032, DOI 10.1007/s00028-003-0503-1
Jérôme Droniou and Cyril Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal. 182 (2006), no. 2, 299–331. MR 2259335, DOI 10.1007/s00205-006-0429-2
Ghorbel A. and Monneau R., Well-posedness and numerical analysis of a one-dimensional non-local transport equation modelling dislocations dynamics, submitted for publication.
Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. MR 1410987, DOI 10.1007/978-1-4612-0713-9
S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81(123) (1970), 228–255 (Russian). MR 267257
B. Jourdain, Probabilistic approximation via spatial derivation of some nonlinear parabolic evolution equations, Monte Carlo and quasi-Monte Carlo methods 2004, Springer, Berlin, 2006, pp. 197–216. MR 2208710, DOI 10.1007/3-540-31186-6_{1}3
Benjamin Jourdain, Sylvie Méléard, and Wojbor A. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli 11 (2005), no. 4, 689–714. MR 2158256, DOI 10.3150/bj/1126126765
Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI 10.1007/BF01762360
S. Salapaka, A. Peirce, and M. Dahleh, Analysis of a circulant based preconditioner for a class of lower rank extracted systems, Numer. Linear Algebra Appl. 12 (2005), no. 1, 9–32. MR 2114602, DOI 10.1002/nla.390
Halil Mete Soner, Optimal control with state-space constraint. II, SIAM J. Control Optim. 24 (1986), no. 6, 1110–1122. MR 861089, DOI 10.1137/0324067
Dan Stanescu, Dongjin Kim, and Wojbor A. Woyczynski, Numerical study of interacting particles approximation for integro-differential equations, J. Comput. Phys. 206 (2005), no. 2, 706–726. MR 2143331, DOI 10.1016/j.jcp.2004.12.023
Charles Van Loan, Computational frameworks for the fast Fourier transform, Frontiers in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153025, DOI 10.1137/1.9781611970999
Wojbor A. Woyczyński, Lévy processes in the physical sciences, Lévy processes, Birkhäuser Boston, Boston, MA, 2001, pp. 241–266. MR 1833700