Numerische Mathematik

, Volume 67, Issue 3, pp 315-344

First online:

Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations

  • Marizio FalconeAffiliated withDipartimento di Matematica, Universit\`a di Roma ``La Sapienza", P.le Aldo Moro, 2, I-00185 Roma, Italy
  • , Roberto FerrettiAffiliated withDipartimento di Matematica, Universit\`a di Roma ``Tor Vergata", v. Fontanile di Carcaricola, I-00133 Roma, Italy

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A general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given. The method is based on a discrete version of the Dynamic Programming Principle. We prove a general convergence result for this class of approximation schemes also obtaining, under more restrictive assumptions, an estimate in\(L^\infty\) of the order of convergence and of the local truncation error. The schemes can be applied, in particular, to the stationary linear first order equation in \({\Bbb R}^n\). We present several examples of schemes belonging to this class and with fast convergence to the solution.

Mathematics Subject Classification (1991): 65N12, 49L20