An anti-diffusive scheme for viability problems

This paper is concerned with the numerical approximation of viability kernels. The method described here provides an alternative approach to the usual viability algorithm. We first consider a characterization of the viability kernel as the value function of a related optimal control problem, and then use a specially relevant numerical scheme for its approximation. Since this value function is discontinuous, usual discretization schemes (such as finite differences) would provide a poor approximation quality because of numerical diffusion. Hence, we investigate the Ultra-Bee scheme, particularly interesting here for its anti-diffusive property in the transport of discontinuous functions. Although currently there is no available convergence proof for this scheme, we observed that numerically, the experiments done on several benchmark problems for computing viability kernels and capture basins are very encouraging compared to the viability algorithm, which fully illustrates the relevance of this scheme for numerical approximation of viability problems. Key-words: Viability kernel, Capture basin, Ultra-Bee scheme, Optimal control.