More on stochastic and variational approach to the Lax-Friedrichs scheme

Soga, Kohei

doi:10.1090/mcom/3061

More on stochastic and variational approach to the Lax-Friedrichs scheme
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Math. Comp. 85 (2016), 2161-2193 Request permission

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.

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