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

Alessio Basti, Fabio Camilli

TL;DR

The paper addresses $L^p$ error estimates for monotone schemes approximating convex Hamilton-Jacobi equations on the $d$-dimensional torus. It leverages the nonlinear adjoint method to obtain an $L^1$ error bound of order $1$ for both finite-difference and semi-Lagrangian discretizations, under standard convexity and regularity assumptions, and then derives $L^p$ estimates for all finite $p>1$ through interpolation with $L^\infty$. The analysis covers semi-discrete and fully discrete schemes, providing a unified framework that extends to a broad class of monotone numerical Hamiltonians and improves existing results. The results have practical impact for robust convergence analysis and error control in numerical approximations of Hamilton-Jacobi equations on periodic domains.

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.

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

TL;DR

The paper addresses error estimates for monotone schemes approximating convex Hamilton-Jacobi equations on the -dimensional torus. It leverages the nonlinear adjoint method to obtain an error bound of order for both finite-difference and semi-Lagrangian discretizations, under standard convexity and regularity assumptions, and then derives estimates for all finite through interpolation with . The analysis covers semi-discrete and fully discrete schemes, providing a unified framework that extends to a broad class of monotone numerical Hamiltonians and improves existing results. The results have practical impact for robust convergence analysis and error control in numerical approximations of Hamilton-Jacobi equations on periodic domains.

Abstract

We establish error estimates for monotone numerical schemes approximating Hamilton-Jacobi equations on the -dimensional torus. Using the adjoint method, we first prove a error bound of order one for finite-difference and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian. By interpolation, we also obtain estimates for every finite . Our analysis covers a broad class of schemes, improves several existing results, and provides a unified framework for discrete error estimates.
Paper Structure (7 sections, 16 theorems, 109 equations)

This paper contains 7 sections, 16 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Assumptions and preliminaries
  3. Estimates for the finite difference scheme
  4. The fully discrete finite difference scheme
  5. The estimates for the semi-Lagrangian scheme
  6. The fully discrete semi-Lagrangian scheme
  7. Proofs of technical lemmas

Key Result

Lemma 1.2

For a function $u:{\mathbb{T}}^d\to \mathbb{R}$, the following assertions are equivalent:

Theorems & Definitions (28)

  • Definition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Remark 2.1
  • Proposition 2.2: Stability and semiconcavity
  • Theorem 2.3: Convergence rate in $L^\infty$
  • Remark 2.4
  • Lemma 2.5: Finite-difference properties
  • Theorem 2.6: $L^1$ error estimate; finite difference scheme
  • proof
  • ...and 18 more