Error Estimate for a Semi-Lagrangian Scheme for Hamilton-Jacobi Equations on Networks
Elisabetta Carlini, Valentina Coscetti, Marco Pozza
TL;DR
This work addresses the numerical approximation of evolutive Hamilton–Jacobi equations on networks and provides a rigorous convergence error estimate for a semi-Lagrangian scheme. By establishing the equivalence of two viscosity-solution notions and leveraging a general convergence result for monotone, stable, and consistent schemes, the authors prove an error bound of order $\frac{1}{2}$ under a time-step restriction. The analysis rests on introducing modified Hamiltonians and a flux-limiter framework to handle vertex junctions, with a discretization that is explicit and amenable to parallelization. Numerical experiments on networks with Hamiltonians independent and dependent on the state variable illustrate the practical validity of the error estimate and reveal that the restriction is sufficient but not necessary for convergence. Overall, the work advances reliable, scalable schemes for HJ on networks and clarifies the role of time-step restrictions in achieving quantified convergence rates.
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).