Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L₂, L_∞ and three invariants I₁, I₂, and I₃ are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.     

Influence of capillary effects on strength of non-saturated porous mediaInfluence des effets capillaires sur la résistance d'un milieu poreux non saturé

The mechanical behavior of a partially saturated porous medium is addressed by means of a micro-to-macro reasoning. First, an estimate of the quadratic average over the solid phase of the equivalent shear strain is proposed. The latter is used in the framework of a nonlinear homogenization technique (‘modified secant’ method) in order to model the nonlinear poroelastic behavior in partially saturated conditions. The determination of the macroscopic strength criterion is then considered. Finally, the influence of membrane tension effects on strength is investigated. To cite this article: L. Dormieux et al., C. R. Mecanique 334 (2006).     

A penalty continuation method for the ∓∞ solution of overdetermined linear systems

A new algorithm for the ∓_∞ solution of overdetermined linear systems is given. The algorithm is based on the application of quadratic penalty functions to a primal linear programming formulation of the ∓_∞ problem. The minimizers of the quadratic penalty function generate piecewise-linear non-interior paths to the set of ∓_∞ solutions. It is shown that the entire set of ∓_∞ solutions is obtained from the paths for sufficiently small values of a scalar parameter. This leads to a finite penalty/continuation algorithm for ∓_∞ problems. The algorithm is implemented and extensively tested using random and function approximation problems. Comparisons with the Barrodale-Phillips simplex based algorithm and the more recent predictor-corrector primal-dual interior point algorithm are given. The results indicate that the new algorithm shows a promising performance on random (non-function approximation) problems.     

Femoral head avascular necrosis in sickle cell anemia: MR characteristics

Since considerable expansion of hematopoietic marrow occurs in patients with sickle cell anemia (SCA), magnetic resonance images of 20 hips in 10 patients with known homozygous SCA were reviewed to determine a) if low signal hematopoietic marrow extended into the femoral capital epiphysis and b) if the MR characteristics of avascular necrosis (AVN) differed depending on the type of epiphyseal marrow. Our results revealed variable epiphyseal marrow type; mixed (fatty and hematopoietic) marrow (42%), fatty marrow (32%), hematopoietic marrow (16%) and hemosiderotic marrow (10%). AVN occurred irrespective of the underlying marrow. Segmental areas of low signal intensity in variable shapes (ring, band, crescent or large homogeneous area) was the most consistent MR manifestation of AVN in SCA. A low signal intensity peripheral rim surrounding a central zone, isointense with epiphyseal marrow on T₁ and T₂ weighted images, was most frequently observed similar to that described in patients without hemoglobinopathy. The notable difference, however, was of segmental areas within the same femoral head that demonstrated variable central zone signal on T₂ weighted images. Further, while an increase in hip joint fluid is commonly seen with both early and advanced AVN in patients without hemoglobinopathy; it was increased in only one hip in patients with SCA. The observed differences in MR characteristics may be due to different pathophysiology of AVN in patients with SCA.     

A mild reduction of aliphatic nitro compounds to imines for further in situ reactions : A simple synthesis of pyrroles

Tributylphosphine-diphenyldisulphide reduces nitroalkanes to imines which can be trapped intramolecularly to give pyrroles.     

An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

This article develops a new two‐level three‐point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u_xx = f(x, t, u, u_x, u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (?u/?x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007