Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

In this paper we study the geometric numerical solution of the so called “good” Boussinesq equation. This goal is achieved by using a convenient space semi‐discretization, able to preserve the corresponding Hamiltonian structure, then using energy‐conserving Runge–Kutta methods in the Hamiltonian boundary value method class for the time integration. Numerical tests are reported, confirming the effectiveness of the proposed method.     

Stable tangential family germs and singularities of their envelopes

A tangential family is a system of regular curves emanating tangentially from another regular curve. We classify tangential family germs which are stable under deformations among tangential families and we study singularities of their envelopes. We also discuss some applications of our results to Differential Geometry. To cite this article: G. Capitanio, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

2-D Numerical Modeling of Bioremediation in Heterogeneous Saturated Soils

This paper presents a numerical solution approach for an existing model for simulating transport and biodegradation in saturated porous media. The discrete approximation makes use of an appropriate blending of mixed-hybrid finite-element and shock-capturing finite-volume schemes. The model is applied for simulating enhanced-bioremediation of highly heterogeneous porous media contaminated by organic pollutants. Injection of water enriched in dissolved oxygen (DO) is considered for accelerating contaminant degradation and concentration of both organic pollutant (substrate) and DO. Heterogeneity is found to produce pools of contaminants which strongly affect DO delivery and, then, the degradation of the organic contaminant. A set of numerical results on representative situations illustrates the effectiveness and the robustness of the present approach. The computational efficiency of the present approach is also estimated in terms of CPU costs and memory requirements.     

Simple tangential family germs and perestroikas of their envelopes

A tangential family is a 1-parameter system of regular curves emanating tangentially from another regular curve. We classify simple tangential family germs up to A-equivalence. We describe perestroikas of envelopes of simple tangential family germs of small codimension under small deformations of the germ among tangential families.     

Null Space Algorithm and Spanning Trees in Solving Darcy's Equation

A Null Space algorithm is considered to solve the augmented system produced by the mixed finite element approximation of Darcy's Law. The method is based on the combination of a LU factorization technique for sparse matrices with an iterative Krylov solver. The computational efficiency of the method relies on the use of spanning trees to compute the LU factorization without fill-in and on a suitable stopping criterion for the iterative solver. We experimentally investigate its performance on a realistic set of selected application problems.This revised version was published online in October 2005 with corrections to the Cover Date.     

Perestroikas of vertex sets at umbilic points

Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane of M at a point p. Those vertices trace out a set, called the vertex set through p. We take p to be a generic umbilic point on M and describe the perestroikas of the vertex set under generic n-parameter small deformations of the surface. One of the consequences of our results is that, in some sense, generically the study of the discriminants of small n-deformations, with n greater than or equal to 2, simplifies to that of 2-deformations. Beyond the Mathematical interest in vertices of families of curves, this work was primarily motivated by the medial representation of shapes in Computer Vision and Image Analysis, where the behavior of vertices plays a crucial role in the qualitative changes of the skeleton or Blum medial axis of curves.     

Continuum modeling of periodic brickwork

Continuum models of periodic masonry brickwork, viewed at a micro-level as a discrete system, are identified within the frame of linearized elasticity. The accuracy of various identification schemes is investigated for standard and micropolar continua, which are directly compared with the help of some numerical benchmarks, for different loading conditions that induce periodic and non-periodic deformation states. It is shown that periodic deformation states of brickwork are exactly reproduced by both continua, provided that a suitable identification scheme is adopted. For non-periodic states micropolar continuum is shown to better reproduce the discrete solutions, due to its capability to take scale effects into account. Both continua are asymptotically equivalent as the characteristic length of the discrete system tends to zero, while providing an upper and a lower bound of the discrete solution.     

A horizontal Chern–Gauss–Bonnet formula on totally geodesic foliations

Annals of Global Analysis and Geometry - Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are...