A Convergent Semi-Lagrangian Scheme for the Game $$\infty $$-Laplacian

We propose a new semi-Lagrangian scheme for the game $$\infty $$ ∞ -Laplacian. We demonstrate the convergence of the scheme to the viscosity solution of the given problem, showing its consistency, monotonicity, and stability. The proof of this result is established following the Barles-Souganidis analysis. This analysis assumes convergence at the boundary in a strong sense and is applied to our proposed scheme, augmented with an artificial viscosity term.