François Bouchon

Numerical methods for the simulation of a corrosion model with moving oxide layer

In this paper, we design numerical methods for a PDE system arising in corrosion modeling. This system describes the evolution of a dense oxide layer. It is based on a drift-diffusion system and includes moving boundary equations. The choice of the numerical methods is justified by a stability analysis and by the study of their numerical performance. Finally, numerical experiments with real-life data shows the efficiency of the developed methods.

The Immersed Interface Technique for Parabolic Problems with Mixed Boundary Conditions

A finite difference scheme is presented for a parabolic problem with mixed boundary conditions. We use an immersed interface technique to discretize the Neumann condition, and we use the Shortley-Weller approximation for the Dirichlet condition. The proof of a discrete maximum principle is given as well as the proof of convergence of the scheme. This convergence is also validated on numerical examples.

Monotonicity of some perturbations of irreducibly diagonally dominant M-matrices

This paper presents a new result concerning the perturbation theory of M-matrices. We give the proof of a theorem showing that some perturbations of irreducibly diagonally dominant M-matrices are monotone, together with an explicit bound of the norm of the perturbation. One of the assumptions concerning the perturbation matrix is that the sum of the entries of each of its row is nonnegative. The resulting matrix is shown to be monotone, although it may not be diagonally dominant and its off diagonal part may have some positive entries. We give as an application the proof of the second order convergence of an non-centered finite difference scheme applied to an elliptic boundary value problem.

An Immersed Interface Technique for the Numerical Solution of the Heat Equation on a Moving Domain

A finite difference scheme for the heat equation with mixed boundary conditions on a moving domain is presented. We use an immersed interface technique to discretize the Neumann condition and the Shortley–Weller approximation for the Dirichlet condition. Monotonicity of the discretized parabolic operator is established. Numerical results illustrate the feasibility of the approach.

A Model Based on Incremental Scales Applied to LES of Turbulent Channel Flow

A model based on incremental scales is applied to LES of incompressible turbulent channel flow. With this approach, the resolved scales are decomposed into large and incremental scales; the incremental scales have a larger (two times) spectral support than the large ones. Both velocity components are advanced in time by integrating their respective equations. At every time step and point in the wall normal direction, the one-dimensional energy spectra of the incremental scales are corrected in order to fit the slopes of the corresponding large scale spectra. LES of turbulent channel flow at two different Reynolds numbers are conducted. Results for both simulations are in good agreement with filtered DNS data. A significant improvement is shown compared to simulations with no model at the same low resolutions as the LES. The computational cost of the incremental method is similar to that of a Galerkin approximation on the same grid.

A Second-Order Immersed Boundary Method for the Numerical Simulation of Two-Dimensional Incompressible Viscous Flows Past Obstacles

We present a new cut-cell method, based on the MAC scheme on Cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. The discretization of the nonlinear terms, written in conservative form, is formulated in the context of finite volume methods. While first order approximations are used in cut-cells the scheme is globally second-order accurate. The linear systems are solved by a direct method based on the capacitance matrix method. Accuracy and efficiency of the method are supported by numerical simulations of 2D flows past a cylinder at Reynolds numbers up to 9,500.

A multilevel method applied to the numerical simulation of two-dimensional incompressible flows past obstacles at high Reynolds number

Numerical simulation of turbulent flows in complex geometries is one of the most investigated fields in computer science in the last decades. But even though the power of supercomputers has regularly increased for many years, it has been understood that the numerical simulation of realistic flows at high Reynolds number would require too many efforts in term of memory and CPU time if one discretizes directly the Navier-Stokes equation.