Abstract: We study a class of monotone numerical schemes for time-dependent Hamilton–Jacobi equations with weak Dirichlet boundary conditions. We get a convergence rate of 1/2 under some usual assumptions on the data, plus an extra assumption on the Hamiltonian H(Du,x) at the boundary ∂Ω. More specifically the mapping p→H(p,x) must satisfy a monotonicity condition for all p in a certain subset of Rⁿ given by Ω. This condition allows the use of the interior subsolution conditions at the boundary in the comparison arguments. We also prove a comparison result and Lipschitz regularity of the exact solution. As an example we construct a Godunov type scheme that can handle the weakened boundary conditions.