Jourdain, B. Probabilistic characteristics method for a one-dimensional inviscid scalar conservation law. (English) Zbl 1013.60022 Ann. Appl. Probab. 12, No. 1, 334-360 (2002). Author’s summary: We are interested in approximating the 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 characteristics 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. Reviewer: Ilya S.Molchanov (Glasgow) Cited in 10 Documents MSC: 60F17 Functional limit theorems; invariance principles 65C35 Stochastic particle methods 35L65 Hyperbolic conservation laws Keywords:conservation law; entropy solution; particle approximation × Cite Format Result Cite Review PDF Full Text: DOI References: [1] BILLINGSLEY, P. (1986). Probability and Measure. Wiley, New York. · Zbl 0649.60001 [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. · Zbl 0962.65075 · doi:10.1515/mcma.1997.3.2.113 [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. · Zbl 0860.60038 · doi:10.1214/aoap/1034968229 [4] BOSSY, M. and TALAY, D. (1997). A stochastic particle method for the Mckean-Vlasov and the Burgers equation. Math. Comp. 66 157-192. JSTOR: · Zbl 0854.60050 · doi:10.1090/S0025-5718-97-00776-X [5] JOURDAIN, B. (1998). Propagation trajectorielle du chaos pour les lois de conservation scalaires. Séminaire de Probabilités 32 215-230. Springer, New York. · Zbl 0913.60094 [6] JOURDAIN, B. (2000). Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab. 2 69-91. · Zbl 0960.60092 · doi:10.1023/A:1010059302049 [7] JOURDAIN, B. (2000). Probabilistic approximation for a porous medium equation. Stochastic Process. Appl. 89 81-99. · Zbl 1054.35067 · doi:10.1016/S0304-4149(00)00014-4 [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. · Zbl 0823.35116 · doi:10.1002/mma.1670161205 [9] PERTHAME, B. and PULVIRENTI, M. (1995). On some large systems of random particles which approximate scalar conservation laws. Asymptotic Anal. 10 263-278. · Zbl 0846.35081 [10] STROOCK, D. W. and VARADHAN, S. R. S. (1997). Multidimensional Diffusion Processes. Springer, New York. · Zbl 1103.60005 [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. · Zbl 0732.60114 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.