The so-called Barles-Souganidis framework [BS91] gives sufficient conditions for convergence of approximation schemes for equations of the form $$F(x,u,Du,D^2u)=0.$$ In particular, it establishes that any approximation scheme that is monotone, stable, and consistent with respect to a limiting equation that satisfies a comparison principle converges to its unique (bounded) viscosity solution.
Their paper is a must-read for anyone performing numerical computations in the field of viscosity solutions.
Caveat: Barles and Souganidis use a stronger notion of comparison principle by accounting for the boundary conditions (sometimes referred to as a strong comparison principle). You can relax their result to more usual notions, but doing so requires that you check convergence at the boundary separately, roughly speaking.
Addendum: when I have time, I will update this answer to include a full proof of the Barles-Souganidis result.