More on stochastic and variational approach to the lax-friedrichs scheme

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2 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)2161-2193
Number of pages33
JournalMathematics of Computation
Volume85
Issue number301
DOIs
Publication statusPublished - 2016

Keywords

  • Calculus of variations
  • Hamilton-Jacobi equation
  • Lax-Friedrichs scheme
  • Random walk
  • Scalar conservation law
  • Weak KAM theory

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics
  • Computational Mathematics

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