$L^p$ Estimates for Numerical Approximation of Hamilton-Jacobi Equations

TL;DR

This paper derives $L^p$ error estimates for monotone numerical schemes solving Hamilton-Jacobi equations, extending previous results and providing a unified approach for error analysis.

Contribution

It introduces a unified framework for $L^p$ error estimates for monotone schemes solving Hamilton-Jacobi equations, improving upon existing results.

Findings

Established $L^1$ error bound of order one for schemes

Derived $L^p$ estimates for all finite $p>1$

Applicable to a broad class of schemes

Abstract

We establish $L^p$ error estimates for monotone numerical schemes approximating Hamilton-Jacobi equations on the $d$-dimensional torus. Using the adjoint method, we first prove a $L^1$ error bound of order one for finite-difference and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian. By interpolation, we also obtain $L^p$ estimates for every finite $p>1$. Our analysis covers a broad class of schemes, improves several existing results, and provides a unified framework for discrete error estimates.