Paper Info
Title
Stochastic and variational approach to the Lax-Friedrichs scheme.
Abstract
A stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws and Hamilton-Jacobi equations generated by space-time dependent flux functions of the Tonelli type was clarified by Soga (2015). The results for the Lax-Friedrichs scheme are extended here to show its time-global stability, the large-time behavior, and error estimates. Also provided is a weak KAM-like theorem for discrete equations that is useful in the numerical analysis and simulation of the weak KAM theory. As one application, a finite difference approximation to effective Hamiltonians and KAM tori is rigorously treated. The proofs essentially rely on the calculus of variations in the Lax-Friedrichs scheme and on the theory of viscosity solutions of Hamilton-Jacobi equations.
Year
DOI
Venue
2015
10.1090/mcom/3061
MATHEMATICS OF COMPUTATION
Keywords
Field
DocType
Lax-Friedrichs scheme,scalar conservation law,Hamilton-Jacobi equation,calculus of variations,random walk,weak KAM theory
Convergence of random variables,Mathematical analysis,Random walk,Calculus of variations,Uniform convergence,Law of large numbers,Stochastic process,Pointwise convergence,Conservation law,Mathematics
Journal
Volume
Issue
ISSN
85
301
0025-5718
Citations 
PageRank 
References 
1
0.41
0
Authors
1
Name
Order
Citations
PageRank
Kohei Soga111.09