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Stochastic and variational approach to the Lax-Friedrichs scheme

Author: Kohei Soga
Journal: Math. Comp. 84 (2015), 629-651
MSC (2010): Primary 65M06, 35L65, 49L25, 60G50
Published electronically: July 22, 2014
MathSciNet review: 3290958
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Abstract: In this paper we present a stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws. This is a finite difference version of Fleming's results ('69) that the vanishing viscosity method is characterized by stochastic processes and calculus of variations. We convert the difference equation into that of the Hamilton-Jacobi type and introduce corresponding calculus of variations with random walks. The stability of the scheme is obtained through the calculus of variations. The convergence of approximation is derived from the law of large numbers in hyperbolic scaling limit of random walks. The main advantages due to our approach are the following: Our framework is basically a.e. pointwise convergence with characterization of ``a.e.'', which yields uniform convergence except ``small'' neighborhoods of shocks; The stability and convergence proofs are verified for arbitrarily large time interval, which are hard to obtain in the case of flux functions of general types depending on both space and time; the approximation of characteristic curves is available as well as that of PDE-solutions, which is particularly important for applications of the Lax-Friedrichs scheme to the weak KAM theory.

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Additional Information

Kohei Soga
Affiliation: Department of Pure and Applied Mathematics, Waseda University, Tokyo 169-8555, Japan
Address at time of publication: CNRS-ENS Lyon, UMPA UMR 5669, 69364 Lyon cedex 7, France

Keywords: Lax-Friedrichs scheme, scalar conservation law, Hamilton-Jacobi equation, calculus of variations, random walk, law of large numbers
Received by editor(s): March 28, 2012
Received by editor(s) in revised form: April 23, 2013, and June 6, 2013
Published electronically: July 22, 2014
Additional Notes: The author was supported by Grant-in-Aid for JSPS Fellows (20-6856)
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.