A semi-Lagrangian scheme for the game $p$-Laplacian via $p$-averaging

TL;DR

This paper introduces a semi-Lagrangian numerical scheme based on p-averages for approximating the game p-Laplacian in two dimensions, demonstrating convergence and accuracy through analysis and numerical tests.

Contribution

The paper proposes a novel semi-Lagrangian scheme for the game p-Laplacian using p-averages, with proven convergence and validated numerical accuracy.

Findings

Scheme converges to viscosity solutions in specific cases

Numerical tests confirm the scheme's accuracy across different p values

The method effectively approximates the game p-Laplacian in 2D

Abstract

We present and analyze an approximation scheme for the two-dimensional game $p$-Laplacian in the framework of viscosity solutions. The approximation is based on a semi-Lagrangian scheme which exploits the idea of $p$-averages. We study the properties of the scheme and prove that it converges, in particular cases, to the viscosity solution of the game $p$-Laplacian. We also present a numerical implementation of the scheme for different values of $p$; the numerical tests show that the scheme is accurate.