Numerical analysis of time-dependent Hamilton–Jacobi equations on networks

A new algorithm for time-dependent Hamilton–Jacobi equations on networks, based on semi-Lagrangian scheme, is proposed. It is based on the definition of viscosity solution for this kind of problems recently given in Siconolfi (J Math Pures Appl (9) 163:702–738, 2022). A thorough convergence analysis, not requiring weak semilimits, is provided. In particular, the check of the supersolution property at the vertices is performed through a dynamical technique which seems new. The scheme is efficient, explicit, allows long time steps, and is suitable to be implemented in a parallel algorithm. We present some numerical tests, showing the advantage in terms of computational cost over the one proposed in Carlini et al. (SIAM J Numer Anal 58:3165–3196, 2020).