Section: New Results
Numerical methods for HJ equations
An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations
Participant : Olivier Bokanowski.
The paper [14] , co-authored with M. Griebel (Fraunhofer SCAI & Univ. Bonn), J. Garcke and I. Klopmpaker (TUB, Berlin) proposes
a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal
with non-linear time-dependent Hamilton-Jacobi Bellman equations.
We focus in particular on front propagation models in higher dimensions which are related to control problems.
We test the numerical efficiency of the method on several benchmark problems up to space dimension
A discontinuous Galerkin scheme for front propagation with obstacles
Participant : Olivier Bokanowski.
In [33] , co-authored with C.-W. Shu (Brown Univ.) and
Y. Cheng (Michigan Univ.), some front propagation problems in the presence of
obstacles are analysed. We extend a previous work (Bokanowski, Cheng and Shu, SIAM J. Scient. Comput., 2011), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation
problems. We follow the formulation of (Bokanowski, Forcadel and Zidani, SIAM J. Control Optim. 2010),
leading to a level set formulation driven by
Semi-Lagrangian discontinuous Galerkin schemes for some first and second order PDEs
Participant : Olivier Bokanowski.
Explicit, unconditionally stable, high order schemes for the approximation of some first and second order linear, time-dependent partial differential equations (PDEs) are proposed in [34] , in collaboration with G. Simarmata (internship 2011, currently in RI dep. of Rabobank). The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin elements. It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010) and of Qiu and Shu (2011), for first order equations, based on exact integration, quadrature rules, and splitting techniques. In particular we obtain high order schemes, unconditionally stable and convergent, in the case of linear second order PDEs with constant coefficients. In the case of non-constant coefficients, we construct "almost" unconditionally stable second order schemes and give precise convergence results. The schemes are tested on several academic examples, including the Black and Scholes PDE in finance.