TY - JOUR

T1 - Stochastic and variational approach to the lax-friedrichs scheme

AU - Soga, Kohei

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2014

Y1 - 2014

N2 - In this paper we present a stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws. This is a finite difference version of Fleming's results ('69) that the vanishing viscosity method is characterized by stochastic processes and calculus of variations. We convert the difference equation into that of the Hamilton-Jacobi type and introduce corresponding calculus of variations with random walks. The stability of the scheme is obtained through the calculus of variations. The convergence of approximation is derived from the law of large numbers in hyperbolic scaling limit of random walks. The main advantages due to our approach are the following: Our framework is basically a.e. pointwise convergence with characterization of "a.e.", which yields uniform convergence except "small" neighborhoods of shocks; The stability and convergence proofs are verified for arbitrarily large time interval, which are hard to obtain in the case of flux functions of general types depending on both space and time; the approximation of characteristic curves is available as well as that of PDE-solutions, which is particularly important for applications of the Lax-Friedrichs scheme to the weak KAM theory.

AB - In this paper we present a stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws. This is a finite difference version of Fleming's results ('69) that the vanishing viscosity method is characterized by stochastic processes and calculus of variations. We convert the difference equation into that of the Hamilton-Jacobi type and introduce corresponding calculus of variations with random walks. The stability of the scheme is obtained through the calculus of variations. The convergence of approximation is derived from the law of large numbers in hyperbolic scaling limit of random walks. The main advantages due to our approach are the following: Our framework is basically a.e. pointwise convergence with characterization of "a.e.", which yields uniform convergence except "small" neighborhoods of shocks; The stability and convergence proofs are verified for arbitrarily large time interval, which are hard to obtain in the case of flux functions of general types depending on both space and time; the approximation of characteristic curves is available as well as that of PDE-solutions, which is particularly important for applications of the Lax-Friedrichs scheme to the weak KAM theory.

KW - Calculus of variations

KW - Hamilton-Jacobi equation

KW - Law of large numbers

KW - Lax-Friedrichs scheme

KW - Random walk

KW - Scalar conservation law

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U2 - 10.1090/S0025-5718-2014-02863-9

DO - 10.1090/S0025-5718-2014-02863-9

M3 - Article

AN - SCOPUS:84896460110

VL - 84

SP - 629

EP - 651

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 292

ER -