Numerical Approximation of the Critical Value of Eikonal Hamilton-Jacobi Equations on Networks

TL;DR

This work studies approximation strategies for the critical value of eikonal equations posed on networks based on the large time behavior of corresponding time-dependent Hamilton-Jacobi equations.

Abstract

The critical value of an eikonal equation is the unique value of a parameter for which the equation admits solutions and is deeply related to the effective Hamiltonian of a corresponding homogenization problem. We study approximation strategies for the critical value of eikonal equations posed on networks. They are based on the large time behavior of corresponding time-dependent Hamilton-Jacobi equations. We provide error estimates and some numerical tests, showing the performance and the convergence properties of the proposed algorithms.

Figures (6)

• Figure 1: Network $\Gamma \subset \mathds{R}^2$: the triangle.
• Figure 2: \ref{['basicalgo']}. Test on the triangle with Hamiltonian dependent on $s$, $A=2/3$, $B=1/2$, $C=1$, $\Delta t=\min_{\gamma\in\mathcal{E}^+}\Delta_\gamma x/12$ and $2000$ fixed iterations.
• Figure 3: \ref{['basicalgo']}. Test on the triangle with Hamiltonian independent of $s$, $A=1$, $B=0$, $C=1$, $\Delta t=(\min_{\gamma\in\mathcal{E}^+}\Delta_\gamma x)/9.1$ and $2000$ fixed iterations.
• Figure 4: Network $\Gamma \subset \mathds{R}^2$: the traffic circle.
• Figure 5: \ref{['basicalgo']}. Test on the traffic circle with Hamiltonian dependent on $s$, $\Delta t=(\min_{\gamma\in\mathcal{E}^+}\Delta_\gamma x)/9.5$ and $2000$ fixed iterations. The third graph refers to the test with $\Delta x=2.50\cdot 10^{-2}$.

Theorems & Definitions (4)

• proof
• proof
• proof : Proof of \ref{['basicerr']}
• proof