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Stochastic and variational approach to finite difference approximation of Hamilton-Jacobi equations

Author: Kohei Soga
Journal: Math. Comp. 89 (2020), 1135-1159
MSC (2010): Primary 65M06, 35F21, 49L25, 60G50
Published electronically: September 10, 2019
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Abstract: Previously, the author presented a stochastic and variational approach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic setting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. 84 (2015), 629-651]. In the current paper, we extend these results to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equations, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks, a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition, and the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and variational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs 5 (1969) 515-530].

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Additional Information

Kohei Soga
Affiliation: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

Keywords: Finite difference scheme, Hamilton-Jacobi equation, viscosity solution, calculus of variations, random walk, law of large numbers
Received by editor(s): March 22, 2018
Received by editor(s) in revised form: February 12, 2019, March 16, 2019, and July 8, 2019
Published electronically: September 10, 2019
Additional Notes: The main part of this work was done, when the author belonged to Unité de mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon, being supported by ANR-12-BS01-0020 WKBHJ as a researcher for academic year 2014–2015, hosted by Albert Fathi. The author was partially supported by JSPS Grant-in-aid for Young Scientists (B) #15K21369
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