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).

Keywords

Error estimate / Hamilton-Jacobi (HJ) equations / Semi-Lagrangian scheme / Embedded networks / 65M15 / 49L25 / 65M12 / 35R02

References

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Funding

MIUR(PRIN Project2022238YY5)

Gruppo Nazionale per il Calcolo Scientifico(CUP_E53C24001950001)

MIUR(PRIN Project2022238YY5)

Avvio alla Ricerca(CUP_B83C24006550001)

European Union - Next Generation EU, Missione 4, Componente 1(CUP_B53C23002010006)

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