Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods

Authors:
M. Falcone;R. Ferretti
Affiliations:
Dipartimento di Matematica, Università di Roma "La Sapienza," P. Aldo Moro, 2, 00185 Roma, Italy;Dipartimento di Matematica, Università di Roma Tre, L.go S. Leonardo Murialdo, 1, 00146 Roma, Italy
Venue:
Journal of Computational Physics
Year:
2002

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Abstract

We study a class of semi-Lagrangian schemes which can be interpreted as a discrete version of the Hopf-Lax-Oleinik representation formula for the exact viscosity solution of first order evolutive Hamilton-Jacobi equations. That interpretation shows that the scheme is potentially accurate to any prescribed order. We discuss how the method can be implemented for convex and coercive Hamiltonians with a particular structure and how this method can be coupled with a discrete Legendre trasform. We also show that in one dimension, the first-order semi-Lagrangian scheme coincides with the integration of the Godunov scheme for the corresponding conservation laws. Several test illustrate the main features of semi-Lagrangian schemes for evolutive Hamilton-Jacobi equations.