The Annals of Applied Probability

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

B. Jourdain

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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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015961167

Digital Object Identifier
doi:10.1214/aoap/1015961167

Mathematical Reviews number (MathSciNet)
MR1890068

Zentralblatt MATH identifier
1013.60022

Subjects
Primary: 65C35: Stochastic particle methods [See also 82C80]
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
Scalar conservation law method of characteristics stochastic particle systems reflected diffusion processes propagation of chaos

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


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