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:

Abstract. This paper is devoted to the numerical investigation of radiative hydrodynamics equations. We focus on non–equilibrium regimes and we design asymptotic preserving schemes which can handle the corresponding stiff equations. Our study includes relativistic effects and Doppler corrections.

In this paper, we are interested in some front propagation problems coming from control problems in d-dimensional spaces, with d ≥ 2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an Hamilton-Jacobi-Bellman equation with discontinuous data, discretized by means of the antidissipative Ultra Bee scheme. We develop an efficient dynamic storage technique suitable for handling front evolutions in large dimension. Then we propose a fast algorithm, showing its relevance on several challenging tests in dimension d = 2, 3, 4. We also compare our method with the techniques usually used in level set methods. Our approach leads to a computational cost as well as a memory allocation scaling as O(Nnb) in most situations, where Nnb is the number of grid nodes around the front. AMS Classification: 65M06, 49L99.

Abstract. The goal of this paper is to study some numerical approximations of particular Hamilton-Jacobi-Bellman equations in dimension 1 and with possibly discontinuous initial data. We investigate two anti-diffusive numerical schemes; the first one is based on the Ultra-Bee scheme, and the second one is based on the Fast Marching Method. We prove the convergence and derive L1-error estimates for both schemes. We also provide numerical examples to validate their accuracy in solving smooth and discontinuous solutions. 1.

In this paper, we review a few aspects of two phase flows where a disperse phase — the particles — interacts with a dense fluid. We are thus led to consider kinetic equations where the leading term is due to the drag force exerted by the fluid on the particles. We discuss several asymptotic questions and present a numerical scheme which is able to treat the multiscale features of the problem. 1

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L 1-error estimates for numerical approximations of

Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous data