Mathematical Models and Methods in Applied Sciences

We analyze a front tracking algorithm for 2×2 systems of conservation laws with non-genuinely nonlinear characteristic fields. The convergence of the corresponding approximate Riemann solvers is established and the basic interaction estimates for the front tracking approximate solutions are provided.

In this paper we deal with the numerical analysis of an optimal control problem of minimax type with finite horizon and final cost. To get numerical approximations we devise here a fully discrete scheme which enables us to compute an approximated solution. We prove that the fully discrete solution converges to the solution of the continuous problem and we also give the order of the convergence rate. Finally we present some numerical results.

In this paper we present a numerical procedure to evaluate the efficiency of finite element numerical methods. We improve some of the ideas proposed in previous works and give a partly theoretical, partly empirical justification in a general framework. The proposed procedure performs an eigenvalue computation, and requires the knowledge of the behavior of the exact operator in order to choose proper norms for the evaluations. In the experiments we focus our attention on the 1-D advection–diffusion problem: we show that our numerical procedure actually gives very sharp indications about the optimality of the tested numerical methods.

In this work we propose a new numerical method for solving a thermomechanical coupled problem which arises as a mathematical model for the evolution of ice sheets profiles and the corresponding temperatures. From the accumulation–ablation ratio, the atmospheric temperature and the geothermic flux, the ice sheet profile is the solution of a moving boundary problem governed by a nonlinear convection–diffusion equation. Moreover, the mathematical model which governs the temperature distribution involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two-phase Stefan formulation of the polythermal regime. The problems are discretized in time with upwind characteristics schemes and in space with finite elements. The nonlinearities are solved either by fixed point methods or by duality techniques. Several numerical simulation examples involving real data sets issued from the Antartic ice sheet are shown.

We consider a two-phase model described by a pressureless gas system with unilateral constraint. We prove weak stability and the existence of weak solutions by passing to the limit in the sticky-blocks dynamics. We obtain the maximum principle on the velocity, the Oleinik entropy condition and local entropy inequalities. Initial data are taken in a very weak sense since the solution can jump initially in time.

We study the asymptotic behavior of the inductance coefficient for a thin toroidal inductor whose thickness depends on a small parameter ε > 0. We give an explicit form of the singular part of the corresponding potential u^ε which allows to construct the limit potential u (as ε → 0) and an approximation of the inductance coefficient L^ε. We establish some estimates of the deviation u^ε - u and of the error of approximation of the inductance. We show that L^ε behaves asymptotically as ln ε, when ε → 0.

A filtering problem when the pair state-observations is a Markov process is analyzed. Extending a Kunita result to our frame, strong uniqueness for the filtering equation is obtained. In the discrete case, a finite state approximation for the filter is considered and an estimate of the approximation error is given.