### 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 language | English |
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Pages (from-to) | 2161-2193 |

Number of pages | 33 |

Journal | Mathematics of Computation |

Volume | 85 |

Issue number | 301 |

DOIs | |

Publication status | Published - 2016 |

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### 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