Asymptotic Analysis - Volume 38, issue 1

Authors: Balabane, Mikhael

Article Type: Research Article

Abstract: In order to compute solutions of the Helmholtz or the Maxwell equations, in the complementary set in $\mathbb{R}$ 3 of a scatterer, one use usually a Domain Decomposition technique when the scatterer is huge. In this paper, we give an alternative to this technique, that we call Boundary Decomposition. We decompose the scatterer itself, into sub‐scatterers, and give the solution as a sum of contributions of each sub‐scatterer. From this point of view, it is an homological type decomposition. It is given here for disjoint sub‐scatterers, and the equations are solved in the frequency domain. Each contribution of …a sub‐scaterer is computed as a sum of a series: an algorithm of Jacobi type is given and prove to converge. The theoretical results needed are set and the proofs for convergence of the algorithm are given. Note that the algorithm does not rely on properties of any specific numerical method for solving the equations. It only relies on properties of the equation itself. So at each step, one can use a specific solver adapted to the geometry of the sub‐scatterer dealt with at that step. This algorithm fits distributed computations. It is now in use in industrial CAD, namely at Aerospatiale‐EADS France.1 Show more

Citation: Asymptotic Analysis, vol. 38, no. 1, pp. 1-10, 2004

Authors: Vakulenko, S. | Volpert, V.

Article Type: Research Article

Abstract: The Cauchy problem for reaction–diffusion systems is studied. A notion of generalized travelling waves (GTW) is introduced to describe the large time behavior of solutions, in particular for the case of a nonhomogeneous medium. If the nonhomogeneous perturbation is asymptotically small, existence and stability of GTW is proved and their velocity is found. If the unperturbed systems is decoupled, the behavior of GTW for the perturbed system can be periodic or even chaotic both for homogeneous and nonhomogeneous perturbations. Dispersion of GTW on nonhomogeneities and applications to combustion problems are discussed.

Keywords: travelling waves, reaction–diffusion systems, attractors

Citation: Asymptotic Analysis, vol. 38, no. 1, pp. 11-33, 2004

Authors: Abidi, H. | Danchin, R.

Article Type: Research Article

Abstract: We consider the inviscid limit of incompressible two‐dimensional fluids with initial vorticity in L∞ and in some Besov space Bη 2,∞ with low regularity index. We obtain a general result of strong convergence in L2 which applies to the case of vortex patches with smooth boundaries. The rate of convergence we find is (νt)3/4 (where ν stands for the viscosity and t, for the time). It improves the (νt)1/2 rate given by P. Constantin and J. Wu in (Nonlinearity 8 (1995), 735–742). Besides, it is shown to be optimal in the case of circular vortex …patches. Show more

Citation: Asymptotic Analysis, vol. 38, no. 1, pp. 35-46, 2004

Authors: Dumas, E.

Article Type: Research Article

Abstract: We give asymptotic descriptions of smooth oscillating solutions of hyperbolic systems with variable coefficients, in the weakly nonlinear diffractive optics regime. The dependence of the coefficients of the system in the space–time variable (corresponding to propagation in a non‐homogeneous medium) implies that the rays are not parallel lines – the same occurs with non‐planar initial phases. Approximations are given by WKB asymptotics with 3‐scales profiles and curved phases. The fastest scale concerns oscillations, while the slowest one describes the modulation of the envelope, which is along rays for the oscillatory components. We consider two kinds of behaviors at the intermediate …scale: ‘weakly decaying’ (Sobolev), giving the transverse evolution of a ‘ray packet’, and ‘shock‐type’ profiles describing a region of rapid transition for the amplitude. Show more

Citation: Asymptotic Analysis, vol. 38, no. 1, pp. 47-91, 2004