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

Abstract We examine the numerical approximation of time-dependent Hamilton-Jacobi (HJ) equations on networks, providing a convergence error estimate for the semi-Lagrangian scheme introduced in Carlini and Siconolfi (Numerical analysis of time-dependent HJ equations on networks. 2023. https://doi.org/10.48550/arXiv.2310.06092 ), where convergence was proven without an error estimate. We derive a convergence error estimate of order one-half. This is achieved by showing the equivalence between two definitions of solutions to this problem proposed in Imbert and Monneau (Ann Sci Éc Norm Supér 50(2): 357–448, 2017) and Siconolfi (J Math Pures Appl 163: 702–738, 2022), a result of independent interest, and applying a general convergence result from Carlini et al. (SIAM J Numer Anal 58(6): 3165–3196, 2020).