Convergence and Rates for Hamilton-Jacobi Equations with Kirchoff Junction Conditions

Abstract: We investigate rates of convergence for two approximation schemes of time-independent and time-dependent Hamilton-Jacobi equ-ations with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton-Jacobi equations, we obtain the classical $\epsilon<sup>{\frac{1}{2}}$ rate, while we obtain an $\epsilon<sup>{\frac{1}{7}}$ rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a quantified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.