## The Annals of Probability

### Rate of convergeance of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval

#### Abstract

In this paper, we give a probabilistic interpretation of a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem. The underlying nonlinear stochastic process is reflected at the boundary to take into account the Dirichlet conditions. After proving uniqueness for the martingale problem, we show existence thanks to a propagation of chaos result. Indeed we exhibit a system of N interacting particles, the empirical measure of which converges to the unique solution of the martingale problem as $N\to+\infty$. As a consequence, the solution of the viscous conservation law can be approximated thanks to a numerical algorithm based on the simulation of the particle system. When this system is discretized in time thanks to the Euler-Lépingle scheme, we show that the rate of convergence of the error is in $\OO(\Delta t +1/\sqrt{N})$, where $\Delta t$ denotes the time step. Finally, we give numerical results which confirm this theoretical rate.

#### Article information

Source
Ann. Probab., Volume 30, Number 4 (2002), 1797-1832.

Dates
First available in Project Euclid: 10 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1039548372

Digital Object Identifier
doi:10.1214/aop/1039548372

Mathematical Reviews number (MathSciNet)
MR1944006

Zentralblatt MATH identifier
1013.60080

#### Citation

Bossy, Mireille; Jourdain, Benjamin. Rate of convergeance of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval. Ann. Probab. 30 (2002), no. 4, 1797--1832. doi:10.1214/aop/1039548372. https://projecteuclid.org/euclid.aop/1039548372

#### References

• [1] BENACHOUR, S., ROy NETTE, B. and VALLOIS, P. (2001). Branching processes associated with 2d-Navier-Stokes equation. Rev. Math. Iberoamericana 17 331-373.
• [2] BENTON, E. and PLATZMAN, G. (1972). A table of solution of the one-dimensional Burgers equation. Quart. Appl. Math. 195-212.
• [3] BOSSY, M. (2000). Optimal rate of convergence of a stochastic particle method to solutions of 1D scalar conservation law equations. Rapport de Recherche RR-3924, INRIA.
• [4] BOSSY, M. and TALAY, D. (1996). Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation. Ann. Appl. Probab. 6 818-861.
• [5] BOSSY, M. and TALAY, D. (1997). A stochastic particle method for the McKean-Vlasov and the Burgers equation. Math. Comp. 66 157-192.
• [6] COSTANTINI, C., PACCHIAROTTI, B. and SARTORETTO, F. (1998). Numerical approximation for functionals of reflecting diffusion processes. SIAM J. Appl. Math. 58 73-102.
• [7] FRIEDMAN, A. (1964). Partial Differential Equations of Parabolic Ty pe. Prentice-Hall, New York.
• [8] JOURDAIN, B. (2000). Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab. 2 69-91.
• [9] LADy ZENSKAJA, O. A., SOLONNIKOV, V. A. and URAL'CEVA, N. N. (1988). Linear and Quasilinear Equations of Parabolic Ty pe. Translations of Mathematical Monographs 23.
• [10] LÉPINGLE, D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comput. Simulation 38 119-126.
• [11] LIONS, P. L. and SZNITMAN, A. S. (1984). Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511-537.
• [12] MÉLÉARD, S. and ROELLY-COPPOLETTA, S. (1987). A propagation of chaos result for a sy stem of particles with moderate interaction. Stochastic Process. Appl. 26 317-332.
• [13] SZNITMAN, A. S. (1984). Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56 311-336.
• [14] SZNITMAN, A. S. (1991). Topics in propagation of chaos. Ecole d'Eté de probabilités de St. Flour XIX. Lecture Notes in Math. 1464. Springer, New York.