Study and implementation of two domain decomposition methods: a one-dimensional approach to molecular-scale detonation initiation.

Summary The work presented here focuses on two distinct topics: in the first part, we deal with two domain decomposition methods inspired by Schwarz's method, on non-overlapping subdomains, for general elliptic problems. These methods are based on the alternative solution of subproblems on the subdomains, with mixed boundary conditions on the interfaces, of Robin for the first method, while for the second one, we introduce an operator acting on the trace term. We demonstrate the convergence of these two methods applied to continuous problems, and establish that they can be interpreted as Peaceman-Rachford methods. After a brief spectral study, we propose general convergence results in the framework of finite difference approximations. We then compare them with those of numerical experiments. The second method, which we can interpret as a preconditioned version of the first one, is more efficient from the continuous point of view, for which we demonstrate the geometric convergence, and from the discrete point of view, for which we establish that the convergence is independent of the discretization step. In the second part, we study a molecular scale detonation initiation problem, modeled by a one-dimensional quantum system perturbed by a potential shock wave. Our goal is to predict the final energy state of the system. We propose the direct numerical integration by a Runge-Kutta method of the Schrodinger equation verified by the wave function of the system decomposed on the basis of the eigenstates. We validate the method for small values of the exciter potential, and conduct some numerical experiments. From the performance analysis we deduce that this method is not powerful enough to be generalized to three-dimensional models, but can be used as a validation tool.