 
Summary: NUMERICAL DISCRETIZATION OF BOUNDARY CONDITIONS FOR FIRST ORDER
HAMILTON{JACOBI EQUATIONS
R
EMI ABGRALL
Abstract. We provide two simple ways of discretizing a large class of boundary conditions for rst order Hamilton Jacobi
equations. We show the convergence of the numerical scheme under mild assumptions. However, many type 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 rst 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.
1. Introduction. The problem of discretizing rst order Hamilton{Jacobi in R N has been considered
by several authors (see e.g. [8, 9, 3]) on various type of meshes (see previous references and [1]). However, up
to our knowledge, the discretization of boundary condition has not yet been considered in a systematic way.
The aim of this paper is to provide a simple and systematic way of discretizing a wide variety of boundary
conditions. This is done in the framework of discontinuous viscosity solutions [4]. More precisely, we consider
the following problem
H(x; u; Du) = 0 x
2
F (x; u; Du) = 0 x 2
