Christine Bernardi

Computing Gowdy spacetimes via spectral evolution in future and past directions

We consider a system of nonlinear wave equations with constraints that arises from the Einstein equations of general relativity and describes the geometry of the so-called Gowdy symmetric spacetimes on T3. We introduce two numerical methods, which are based on the pseudo-spectral approximation. The first approach relies on marching in the future timelike direction and toward the singularity t = 0 and is based on a novel nonlinear transformation, which allows us to reduce the nonlinear source terms to simple quadratic products of the unknown variables. The second approach is designed from asymptotic formulae that are available near this singularity, and evolves the solutions in the past timelike direction from the final data given at t = 0. Numerical experiments are presented in various regimes, including cases where spiky structures are observed as the singularity is approached. The proposed backward strategy leads to a robust numerical method which allows us to accurately simulate the long-time behavior of a large class of Gowdy spacetimes.

Derivative with respect to discontinuities in the porosity

We investigate the sensitivity of hydrostatic pressure of flows through porous media with respect to the position of the soil layers. Indeed, these induce discontinuities of the porosity which is a piecewise constant coefficient κ of the partial differential equation satisfied by the pressure u and it leads to the computation of the derivative of u with respect to changes in position of discontinuity surfaces of κ. The analysis relies on a mixed formulation of the problem. Preliminary numerical simulations are given to illustrate the theory. An applica- tion to a simple inverse problem is also given. c 2002 Academie des sciences/ ´ Editions scientifiques et medicales Elsevier SAS D´´

Spectral discretization of a mixed Schro¨dinger boundary-value problem and application

Abstract Paraxial approximation to the Helmholtz equation in ocean acoustic leads to solve a mixed Schrodinger boundary-value problem. The numerical analysis is performed combining a spectral method in the depth direction and a leap-frog scheme in the propagation direction. A convergence analysis for the approximation is developed, providing error estimates.