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Numerical Discretization of Boundary Conditions for First Order Hamilton-Jacobi Equations

We provide two simple ways of discretizing a large class of boundary conditions for first order Hamilton--Jacobi 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.