The so-called *Barles-Souganidis framework* [\[BS91\]][1] 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.

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**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.

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**Addendum**: when I have time, I will update this answer to include a full proof of the Barles-Souganidis result.

  [1]: http://content.iospress.com/articles/asymptotic-analysis/asy4-3-05