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NUMERICAL DISCRETIZATION OF BOUNDARY CONDITIONS FOR FIRST ORDER HAMILTONJACOBI EQUATIONS
 

Summary: NUMERICAL DISCRETIZATION OF BOUNDARY CONDITIONS
FOR FIRST ORDER HAMILTON­JACOBI EQUATIONS
R´EMI ABGRALL
SIAM J. NUMER. ANAL. c 2003 Society for Industrial and Applied Mathematics
Vol. 41, No. 6, pp. 2233­2261
Abstract. We provide two simple ways of discretizing a large class of boundary conditions for
first order Hamilton­Jacobi equations. We show the convergence of the numerical scheme under mild
assumptions. However, many types of such boundary conditions can be written in this way. Some
provide "good" numerical results (i.e., without boundary layers), whereas others do not. To select
a good one, we first give some general results for monotone schemes which mimic the maximum
principle of the continuous case, and then we show in particular cases that no boundary layer can
exist. Some numerical applications illustrate the method. An extension to a geophysical problem is
also considered.
Key words. Hamilton­Jacobi equations, approximation of boundary conditions
AMS subject classifications. 65M60, 65N99, 35L60, 65Z05, 35Q58
DOI. 10.1137/S0036142998345980
1. Introduction. The problem of discretizing first order Hamilton­Jacobi equa-
tions in RN
has been considered by several authors (see, e.g., [8, 9, 3]) on various types
of meshes (see the previous references and [1]). However, in our knowledge, the dis-

  

Source: Abgrall, Rémi - Institut de Mathematiques de Bordeaux, Université Bordeaux

 

Collections: Mathematics