Tamtam Proceedings - lamsin

A method for optimal control problems 81White cells in the graphs represent null values, dark gray cells represent values between0 and 1 and light gray cells represent value 1. In this example the adaptative gridhas about 1000 cells whereas a regular equivalent grid would have 4096 cells: the gain is4 when using an adptative grid. But this gain is more important when we choose a highermaximal level of refinement. For instance it is of the order 100 for a maximal level=10.Note also that the discontinuity of the exact solution and the one of the numerical solutionlie in the same cell: the numerical solution localises the right cell where the discontinuitylies: this is the antidissipative behavior of the scheme.5. References[1] M.BARDI, I.CAPUZZO-DOLCETTA, “Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations.”, Systems and Control: Foundations and Applications.Birkhäuser,Boston, 1997.[2] G.BARLES, “Solutions de viscosité des équations de Hamilton-Jacobi.”, Mathématiques etapplications. Springer, Paris,17, 1994.[3] O.BOKANOWSKI, H.ZIDANI, “Anti-dissipative schemes for advection and application toHamilton-Jacobi-Bellmann equations.”, Preprint, INRIA Report RR-5337, 2004.[4] B.DÉSPRÈS, F.LAGOUTIÈRE, “ Contact discontinuity capturing schemes for linear advectionand compressible gas dynamics.”, J.Sci. Comput. 16(2001), 479-524.[5] I.GARGANTINI, “ An effective way to represent quadtrees.”, Communications of the ACM,25-12(1982), 905-910.TAMTAM –Tunis– 2005