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A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method
Author(s) -
Anguelov Roumen,
Lubuma Jean M.S.,
Minani Froduald
Publication year - 2009
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1148
Subject(s) - mathematics , discretization , monotone polygon , finite element method , hamilton–jacobi equation , convergence (economics) , finite difference coefficient , scheme (mathematics) , finite difference , mathematical analysis , mixed finite element method , strongly monotone , rate of convergence , finite difference method , space (punctuation) , geometry , channel (broadcasting) , linguistics , physics , philosophy , engineering , electrical engineering , economics , thermodynamics , economic growth
Abstract A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.