{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:13:16Z","timestamp":1776863596554,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"301","license":[{"start":{"date-parts":[[2017,2,10]],"date-time":"2017-02-10T00:00:00Z","timestamp":1486684800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>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.<\/p>","DOI":"10.1090\/mcom\/3061","type":"journal-article","created":{"date-parts":[[2016,2,10]],"date-time":"2016-02-10T11:44:22Z","timestamp":1455104662000},"page":"2161-2193","source":"Crossref","is-referenced-by-count":6,"title":["More on stochastic and variational approach to the Lax-Friedrichs scheme"],"prefix":"10.1090","volume":"85","author":[{"given":"Kohei","family":"Soga","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,2,10]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1088\/0951-7715\/18\/1\/006","article-title":"The asymptotic behaviour of solutions of the forced Burgers equation on the circle","volume":"18","author":"Bernard, Patrick","year":"2005","journal-title":"Nonlinearity","ISSN":"https:\/\/id.crossref.org\/issn\/0951-7715","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s00220-002-0781-5","article-title":"Aubry-Mather theory and Hamilton-Jacobi equations","volume":"235","author":"Bessi, Ugo","year":"2003","journal-title":"Comm. 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