Error Estimate for a Semi-Lagrangian Scheme for Hamilton-Jacobi Equations on Networks

TL;DR

This paper provides a convergence error estimate of order one-half for a semi-Lagrangian scheme solving time-dependent Hamilton-Jacobi equations on networks, building on previous convergence proofs without error bounds.

Contribution

It introduces an explicit error estimate for the semi-Lagrangian scheme, establishing equivalence between solution definitions and applying a general convergence result.

Findings

Error estimate of order one-half for the scheme

Equivalence of solution definitions on networks

Application of a general convergence theorem

Abstract

We examine the numerical approximation of time-dependent Hamilton-Jacobi equations on networks, providing a convergence error estimate for the semi-Lagrangian scheme introduced in (Carlini and Siconolfi, 2023), where convergence was proven without an error estimate. We derive a convergence error estimate of order one-half. This is achieved showing the equivalence between two definitions of solutions to this problem proposed in (Imbert and Monneau, 2017) and (Siconolfi, 2022), a result of independent interest, and applying a general convergence result from (Carlini, Festa and Forcadel, 2020).