Numerical Analysis of time-dependent Hamilton-Jacobi Equations on Networks

TL;DR

This paper introduces a new semi-Lagrangian numerical scheme for solving time-dependent Hamilton-Jacobi equations on networks, with proven convergence and efficient implementation advantages.

Contribution

It proposes a novel algorithm based on viscosity solutions for Hamilton-Jacobi equations on networks, including a new technique for verifying supersolution properties at vertices.

Findings

The scheme is explicit and allows long time steps.

It is computationally more efficient than previous methods.

Numerical tests demonstrate its effectiveness and efficiency.

Abstract

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. 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 [7]