Convergence Rate of Monotone Numerical Schemes for Hamilton–Jacobi Equations with Weak Boundary Conditions

Abstract

We study a class of monotone numerical schemes for time-dependent Hamilton–Jacobi equations with weak Dirichlet boundary conditions. We get a convergence rate of $\frac{1}{2}$ under some usual assumptions on the data, plus an extra assumption on the Hamiltonian $H(Du,x)$ at the boundary $\partial\Omega$. More specifically the mapping $p\to H(p,x)$ must satisfy a monotonicity condition for all p in a certain subset of $\mathbf{R}^n$ given by $\Omega$. This condition allows the use of the interior subsolution conditions at the boundary in the comparison arguments. We also prove a comparison result and Lipschitz regularity of the exact solution. As an example we construct a Godunov-type scheme that can handle the weakened boundary conditions.