## More on stochastic and variational approach to the Lax-Friedrichs scheme

HTML articles powered by AMS MathViewer

- by Kohei Soga PDF
- Math. Comp.
**85**(2016), 2161-2193 Request permission

## Abstract:

A stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws and Hamilton-Jacobi equations generated by space-time dependent flux functions of the Tonelli type was clarified by Soga (2015). The results for the Lax-Friedrichs scheme are extended here to show its time-global stability, the large-time behavior, and error estimates. Also provided is a weak KAM-like theorem for discrete equations that is useful in the numerical analysis and simulation of the weak KAM theory. As one application, a finite difference approximation to effective Hamiltonians and KAM tori is rigorously treated. The proofs essentially rely on the calculus of variations in the Lax-Friedrichs scheme and on the theory of viscosity solutions of Hamilton-Jacobi equations.## References

- Patrick Bernard,
*The asymptotic behaviour of solutions of the forced Burgers equation on the circle*, Nonlinearity**18**(2005), no. 1, 101–124. MR**2109469**, DOI 10.1088/0951-7715/18/1/006 - Ugo Bessi,
*Aubry-Mather theory and Hamilton-Jacobi equations*, Comm. Math. Phys.**235**(2003), no. 3, 495–511. MR**1974512**, DOI 10.1007/s00220-002-0781-5 - Jean Bourgain, François Golse, and Bernt Wennberg,
*On the distribution of free path lengths for the periodic Lorentz gas*, Comm. Math. Phys.**190**(1998), no. 3, 491–508. MR**1600299**, DOI 10.1007/s002200050249 - Renato Calleja and Rafael de la Llave,
*A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification*, Nonlinearity**23**(2010), no. 9, 2029–2058. MR**2672635**, DOI 10.1088/0951-7715/23/9/001 - Piermarco Cannarsa and Carlo Sinestrari,
*Semiconcave functions, Hamilton-Jacobi equations, and optimal control*, Progress in Nonlinear Differential Equations and their Applications, vol. 58, Birkhäuser Boston, Inc., Boston, MA, 2004. MR**2041617** - Michael G. Crandall and Andrew Majda,
*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, DOI 10.1090/S0025-5718-1980-0551288-3 - H. S. Dumas,
*Ergodization rates for linear flow on the torus*, J. Dynam. Differential Equations**3**(1991), no. 4, 593–610. MR**1129562**, DOI 10.1007/BF01049101 - Weinan E,
*Aubry-Mather theory and periodic solutions of the forced Burgers equation*, Comm. Pure Appl. Math.**52**(1999), no. 7, 811–828. MR**1682812**, DOI 10.1002/(SICI)1097-0312(199907)52:7<811::AID-CPA2>3.0.CO;2-D - Albert Fathi,
*Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens*, C. R. Acad. Sci. Paris Sér. I Math.**324**(1997), no. 9, 1043–1046 (French, with English and French summaries). MR**1451248**, DOI 10.1016/S0764-4442(97)87883-4 - A. Fathi,
*Weak KAM Theorem in Lagrangian Dynamics*, Cambridge Univ. Press (2011). - Wendell H. Fleming,
*The Cauchy problem for a nonlinear first order partial differential equation*, J. Differential Equations**5**(1969), 515–530. MR**235269**, DOI 10.1016/0022-0396(69)90091-6 - Renato Iturriaga,
*Minimizing measures for time-dependent Lagrangians*, Proc. London Math. Soc. (3)**73**(1996), no. 1, 216–240. MR**1387088**, DOI 10.1112/plms/s3-73.1.216 - H. R. Jauslin, H. O. Kreiss, and J. Moser,
*On the forced Burgers equation with periodic boundary conditions*, Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 133–153. MR**1662751**, DOI 10.1090/pspum/065/1662751 - N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comp. Math. and Math. Phys. 16 (1976), 105-119.
- John N. Mather,
*Action minimizing invariant measures for positive definite Lagrangian systems*, Math. Z.**207**(1991), no. 2, 169–207. MR**1109661**, DOI 10.1007/BF02571383 - Jürgen Moser,
*Recent developments in the theory of Hamiltonian systems*, SIAM Rev.**28**(1986), no. 4, 459–485. MR**867679**, DOI 10.1137/1028153 - Takaaki Nishida and Kohei Soga,
*Difference approximation to Aubry-Mather sets of the forced Burgers equation*, Nonlinearity**25**(2012), no. 9, 2401–2422. MR**2967111**, DOI 10.1088/0951-7715/25/9/2401 - O. A. Oleĭnik,
*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737**, DOI 10.1090/trans2/026/05 - Florin Şabac,
*The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws*, SIAM J. Numer. Anal.**34**(1997), no. 6, 2306–2318. MR**1480382**, DOI 10.1137/S003614299529347X - M. B. Sevryuk,
*KAM-stable Hamiltonians*, J. Dynam. Control Systems**1**(1995), no. 3, 351–366. MR**1354540**, DOI 10.1007/BF02269374 - Kohei Soga,
*Space-time continuous limit of random walks with hyperbolic scaling*, Nonlinear Anal.**102**(2014), 264–271. MR**3182814**, DOI 10.1016/j.na.2014.02.012 - Kohei Soga,
*Stochastic and variational approach to the Lax-Friedrichs scheme*, Math. Comp.**84**(2015), no. 292, 629–651. MR**3290958**, DOI 10.1090/S0025-5718-2014-02863-9 - K. Soga,
*Selection problems of $\mathbb {Z}^2$-periodic entropy solutions and viscosity solutions*, preprint ( arXiv:1501.03594). - Eitan Tadmor,
*The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme*, Math. Comp.**43**(1984), no. 168, 353–368. MR**758188**, DOI 10.1090/S0025-5718-1984-0758188-8

## Additional Information

**Kohei Soga**- Affiliation: Unité de mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France
- Address at time of publication: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
- MR Author ID: 909684
- Email: soga@math.keio.ac.jp
- Received by editor(s): September 17, 2013
- Received by editor(s) in revised form: April 21, 2014, and October 7, 2014
- Published electronically: February 10, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**85**(2016), 2161-2193 - MSC (2010): Primary 65M06, 35L65, 49L25, 60G50, 37J50
- DOI: https://doi.org/10.1090/mcom/3061
- MathSciNet review: 3511278