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

Authors:
F. Cagnetti;D. Gomes;H. V. Tran
Affiliations:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal;Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal and King Abdullah University of Science and Technology (KAUST), CSMSE Division, Thuwal ...;Department of Mathematics, University of California Berkeley, CA 94720-3840, United States
Venue:
Applied Numerical Mathematics
Year:
2013

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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(h) convergence rate in terms of the L^~ 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.