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).

翻译：本文研究网络上时间相关 Hamilton-Jacobi 方程的数值逼近问题，针对 (Carlini and Siconolfi, 2023) 中提出的半拉格朗日格式（该文已证明其收敛性但未给出误差估计），我们给出了收敛误差估计。我们推导出收敛误差的阶为二分之一。这一结果的实现依赖于证明 (Imbert and Monneau, 2017) 与 (Siconolfi, 2022) 中针对该问题提出的两种解定义的等价性（该结果本身具有独立的研究意义），并应用了 (Carlini, Festa and Forcadel, 2020) 中的一般收敛性结论。