# Convergence of a semi-discretization scheme for the Hamilton-Jacobi equation: a new approach with the adjoint method

created by cagnetti on 02 Jun 2011
modified on 13 Nov 2013

[BibTeX]

Published Paper

Inserted: 2 jun 2011
Last Updated: 13 nov 2013

Journal: Appl. Numer. Math.
Volume: 73
Pages: 2-15
Year: 2013

Abstract:

We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L. C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the $O(\sqrt h)$ convergence rate in terms of the $L^\infty$ norm and $O(h)$ in terms of the $L^1$ norm, where $h$ is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper.