A Modification of the Shishkin Discretization Mesh

In this paper we consider a modification of the Shishkin discretization mesh designed for the numerical solution of one-dimensional linear convection-diffusion singularly perturbed problems. The modification consists of a slightly different choice of the transition point between the fine and coarse parts of the mesh. This leads to a better layer-resolving mesh and to an improvement in the accuracy of the computed solution although the convergence order remains the same. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

In situ electrochemical X-ray absorption spectroscopy of oxygen reduction electrocatalysis with high oxygen flux

An in situ electrochemical X-ray absorption spectroscopy (XAS) cell has been fabricated that enables high oxygen flux to the working electrode by utilizing a thin poly(dimethylsiloxane) (PDMS) window. This cell design enables in situ XAS investigations of the oxygen reduction reaction (ORR) at high operating current densities greater than 1 mA in an oxygen-purged environment. When the cell was used to study the ORR for a Pt on carbon electrocatalyst, the data revealed a progressive evolution of the electronic structure of the metal clusters that is both potential-dependent and strongly current-dependent. The trends establish a direct correlation to d-state occupancies that directly tracks the character of the Pt-O bonding present.     

The layer-resolving transformation and mesh generation for quasilinear singular perturbation problems

The relationship is analyzed between layer-resolving transformations and mesh-generating functions for numerical solution of singularly perturbed boundary-value problems. The analysis is carried out for one-dimensional quasilinear problems without turning points, which are discretized by first-order finite-difference schemes. It is proved that if a general layer-resolving function is used to generate the discretization mesh, then the numerical solution converges uniformly in the perturbation parameter.     

The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems

Numerical Algorithms - A linear two-dimensional singularly perturbed convection-diffusion boundary-value problem is considered. The problem is discretized by the upwind finite-difference method....     

The Hermite Scheme for Semilinear Singular Perturbation Problems

A numerical method for singularly perturbed semilinear boundary value problems is given. The method uses the fourth order Hermite scheme on a special discretization mesh. Its stability and convergence are investigated in the discrete $L^1$ norm.     

Synthesis and characterization of [Ir(1,5-cyclooctadiene)(μ-H)]4: a tetrametallic Ir4H4-core, coordinatively unsaturated cluster

Reported herein is the synthesis of the previously unknown [Ir(1,5-COD)(μ-H)](4) (where 1,5-COD = 1,5-cyclooctadiene), from commercially available [Ir(1,5-COD)Cl](2) and LiBEt(3)H in the presence of excess 1,5-COD in 78% initial, and 55% recrystallized, yield plus its unequivocal characterization via single-crystal X-ray diffraction (XRD), X-ray absorption fine structure (XAFS) spectroscopy, electrospray/atmospheric pressure chemical ionization mass spectrometry (ESI-MS), and UV-vis, IR, and nuclear magnetic resonance (NMR) spectroscopies. The resultant product parallels--but the successful synthesis is different from, vide infra--that of the known and valuable Rh congener precatalyst and synthon, [Rh(1,5-COD)(μ-H)](4). Extensive characterization reveals that a black crystal of [Ir(1,5-COD)(μ-H)](4) is composed of a distorted tetrahedral, D(2d) symmetry Ir(4) core with two long [2.90728(17) and 2.91138(17) ?] and four short Ir-Ir [2.78680 (12)-2.78798(12) ?] bond distances. One 1,5-COD and two edge-bridging hydrides are bound to each Ir atom; the Ir-H-Ir span the shorter Ir-Ir bond distances. XAFS provides excellent agreement with the XRD-obtained Ir(4)-core structure, results which provide both considerable confidence in the XAFS methodology and set the stage for future XAFS in applications employing this Ir(4)H(4) and related tetranuclear clusters. The [Ir(1,5-COD)(μ-H)](4) complex is of interest for at least five reasons, as detailed in the Conclusions section.     

Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems

The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the boundary layers.The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed.For the central scheme,error estimates are derived in a discrete L~1 norm.They are of second order and decrease together with the perturbation parameterε.The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically.Numerical results showε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.     

A priori meshes for singularly perturbed quasilinear two-point boundary value problems

A quasilinear singularly perturbed boundary value problem withoutturning points is used as a model problem to analyse and comparethe Bakhvalov and Shishkin discretization meshes. The Shishkinmeshes are generalized and improved. Received 26 October 1998. Accepted 3 June 1999.