## The Annals of Applied Probability

### Probabilistic Characteristics Method for a One-Dimensional Inviscid Scalar Conservation Law

B. Jourdain

#### Abstract

I this paper, we are interested in approxximation th entropy solution of a one-dimensional inviscid scalar conservation law starting from an initial condition with bounded variation owing to a system of interacting diffusions. We modify the system of signed particles associated with the parabolic equation obtained from the addition of a viscous term to this equation by killing couples of particles with opposite sign that merge. The sample paths of the corresponding reordered particles can be seen as probabilistic characteristic along which the approximate solution is constant. This enables us to prove that when the viscosity vanishes as the initial number of particles goes to $+\infty$, the approximate solution converges to the unique entropy solution of the inviscid conservation law. We illustrate this convergence by numerical results.

#### Article information

Source
Ann. Appl. Probab., Volume 12, Number 1 (2002), 334-360.

Dates
First available in Project Euclid: 12 March 2002

https://projecteuclid.org/euclid.aoap/1015961167

Digital Object Identifier
doi:10.1214/aoap/1015961167

Mathematical Reviews number (MathSciNet)
MR1890068

Zentralblatt MATH identifier
1013.60022

#### Citation

Jourdain, B. Probabilistic Characteristics Method for a One-Dimensional Inviscid Scalar Conservation Law. Ann. Appl. Probab. 12 (2002), no. 1, 334--360. doi:10.1214/aoap/1015961167. https://projecteuclid.org/euclid.aoap/1015961167

#### References

• [1] BILLINGSLEY, P. (1986). Probability and Measure. Wiley, New York.
• [2] BOSSY, M., FEZOUI, L. and PIPERNO, S. (1997). Comparison of a stochastic particle method and a finite volume deterministic method applied to Burgers equation. Monte Carlo Methods Appl. 3 113-140.
• [3] 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.
• [4] BOSSY, M. and TALAY, D. (1997). A stochastic particle method for the Mckean-Vlasov and the Burgers equation. Math. Comp. 66 157-192.
• [5] JOURDAIN, B. (1998). Propagation trajectorielle du chaos pour les lois de conservation scalaires. Séminaire de Probabilités 32 215-230. Springer, New York.
• [6] JOURDAIN, B. (2000). Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab. 2 69-91.
• [7] JOURDAIN, B. (2000). Probabilistic approximation for a porous medium equation. Stochastic Process. Appl. 89 81-99.
• [8] KUNIK, M. (1993). A solution formula for a non-convex scalar hyperbolic conservation law with monotone initial data. Math. Methods Appl. Sci. 16 895-902.
• [9] PERTHAME, B. and PULVIRENTI, M. (1995). On some large systems of random particles which approximate scalar conservation laws. Asymptotic Anal. 10 263-278.
• [10] STROOCK, D. W. and VARADHAN, S. R. S. (1997). Multidimensional Diffusion Processes. Springer, New York.
• [11] SZNITMAN, A. S. (1991). Topics in propagation of chaos. Ecole d'Été de Probabilités de Saint-Flour XIX. Lecture Notes in Math. 1464. Springer, New York.