On the principal axes of diffusion in difference schemes for 2d transport problems

Via Taylor series, we associate with a difference stencil L^h approximating Lu := au_x + bu_y its modified equation:

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. By rotating axes to eliminate the 2Bu_xy term, the principle axes through which the diffusion in L^h acts is calculated. Interestingly, for many schemes proposed for 2D transport problems these axes have little to do with the “streamline” and ”crosswind” directions of the continuous problem. Several examples are considered from this point of view.     

Comparison of side-on and end-on coordination of E2 ligands in complexes [W(CO)5E2] (E=N,P, As,Sb, Bi,Si-, Ge-, Sn-, Pb-)

Complexes of W(CO)(5) with neutral diatomic pnictogen ligands N(2), P(2), As(2), Sb(2), and Bi(2) and anionic Group 14 ligands Si(2) (2-), Ge(2) (2-), Sn(2) (2-), and Pb(2) (2-) coordinated in both side-on and end-on fashion have been optimized by using density functional theory at the BP86 level with valence sets of TZP quality. The calculated bond energies have been used to compare the preferential binding modes of each respective ligand. The results were interpreted by analyzing the nature of the interaction between the ligands and the metal fragment using an energy partitioning method. This yields quantitative information regarding the strength of covalent and electrostatic interactions between the metal and ligand, as well as the contributions by orbitals of different symmetry to the covalent bonding. Results show that all the ligands studied bind preferentially in a side-on coordination mode, with the exception of N(2), which prefers to coordinate in an end-on mode. The preference of the heavier homologues P(2)-Bi(2) for binding in a side-on mode over the end-on mode in the neutral complexes [(CO)(5)WE(2)] comes mainly from the much stronger electrostatic attraction in the former species. The energy difference between the side-on and end-on isomers of the negatively charged complexes with the ligands Si(2) (2-), Ge(2) (2-), Sn(2) (2-), and Pb(2) (2-) is much less and it cannot be ascribed to a particular bonding component.     

Analysis of inorganic mercury species associated with airborne particulate matter/aerosols: method development

This paper describes a method for speciation of Hg associated with airborne particulate matter. This method uses a mini-sampler for sample collection and analysis, thermal desorption for separating Hg species, and inductively coupled plasma mass spectrometry (ICP–MS) for identification and quantification of Hg. Coal fly ash spiked with different Hg compounds (e.g. Hg⁰, HgCl₂, HgO, and HgS) was used for qualitative calibration. A standard reference material with a certified value for Hg concentration was used to evaluate the method. When the temperature of the furnace was programmed at a linear rate of increase of 50° min^–1, different Hg compounds could clearly be separated. Three airborne particulate matter samples were collected in parallel in Toronto, ON, Canada and analyzed using this method. Reproducible results were obtained and Hg⁰, HgCl₂, HgO, and HgS species from these samples were detected.     

The nature of the chemical bond revisited. An energy partitioning analysis of diatomic molecules E2 (E=N–Bi, F–I), CO and BF

The nature of the chemical bonds in the diatomic molecules E₂ (E=N–Bi, F–I), CO and BF has been studied with an energy partitioning analysis using gradient-corrected density functional theory calculations. The results make it possible to estimate quantitatively the strength of covalent and electrostatic attractions and the Pauli repulsion between the atoms. The data suggest that some traditional explanations regarding the strength of the molecules should be modified. The energy partitioning analysis shows that the chemical bonds in the group 15 diatomic molecules have significant electrostatic character, which increases from 30.1% in N₂ to 58.3% in Bi₂. The contribution of the electrostatic attraction to the binding interactions in Sb₂ and Bi₂ is larger than the covalent bonding. The strength of the bonding in the triply bonded dinitrogen is less than that of the bonding. The calculations indicate that E is between 32.2% (Bi₂) and 40.0% (P₂) of the total orbital interaction energy (E_orb). The much stronger bond of N₂, as compared with the heavier group 15 E₂ homologues, is not caused by a particularly strong contribution by the bonding, but rather by the relatively large interactions. The comparison of N₂ with isoelectronic CO shows that the electrostatic character in the heteroatomic molecule is slightly smaller (28.8%) than in the homoatomic molecule. The contribution of the bonding in CO is larger (49.2%) than in N₂ (34.3%). The reason why CO has a stronger bond than N₂ is the significantly weaker Pauli repulsion in CO. The electrostatic character of the bonding in BF is slightly larger (32.0%) than in CO and N₂. BF has much weaker -bonding contributions that provide only 11.2% of the covalent interactions, which is why BF has a much weaker bond than CO and N₂. The chemical bonds in the dihalogen molecules have much higher covalent than electrostatic character. The E_orb term contributes between 74.4% (Br₂) and 79.7% (F₂) to the total attractive interactions. The relatively weak bond in F₂ comes from the rather large Pauli repulsion.Contribution to the Jacopo Tomasi Honorary Issue     

Numerical analysis and computational testing of a high accuracy Leray‐deconvolution model of turbulence

We study a computationally attractive algorithm (based on an extrapolated Crank‐Nicolson method) for a recently proposed family of high accuracy turbulence models, the Leray‐deconvolution family. First we prove convergence of the algorithm to the solution of the Navier‐Stokes equations and delineate its (optimal) accuracy. Numerical experiments are presented which confirm the convergence theory. Our 3d experiments also give a careful comparison of various related approaches. They show the combination of the Leray‐deconvolution regularization with the extrapolated Crank‐Nicolson method can be more accurate at higher Reynolds number that the classical extrapolated trapezoidal method of Baker (Report, Harvard University, 1976). We also show the higher order Leray‐deconvolution models (e.g. N = 1,2,3) have greater accuracy than the N = 0 case of the Leray‐α model. Numerical experiments for the 2d step problem are also successfully investigated. Around the critical Reynolds number, the low order models inhibit vortex shedding behind the step. The higher order models, correctly, do not. To estimate the complexity of using Leray‐deconvolution models for turbulent flow simulations we estimate the models' microscale.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Accurate computation of Stokes flow driven by an open immersed interface

We present numerical methods for computing two-dimensional Stokes flow driven by forces singularly supported along an open, immersed interface. Two second-order accurate methods are developed: one for accurately evaluating boundary integral solutions at a point, and another for computing Stokes solution values on a rectangular mesh. We first describe a method for computing singular or nearly singular integrals, such as a double layer potential due to sources on a curve in the plane, evaluated at a point on or near the curve. To improve accuracy of the numerical quadrature, we add corrections for the errors arising from discretization, which are found by asymptotic analysis. When used to solve the Stokes equations with sources on an open, immersed interface, the method generates second-order approximations, for both the pressure and the velocity, and preserves the jumps in the solutions and their derivatives across the boundary. We then combine the method with a mesh-based solver to yield a hybrid method for computing Stokes solutions at N² grid points on a rectangular grid. Numerical results are presented which exhibit second-order accuracy. To demonstrate the applicability of the method, we use the method to simulate fluid dynamics induced by the beating motion of a cilium. The method preserves the sharp jumps in the Stokes solution and their derivatives across the immersed boundary. Model results illustrate the distinct hydrodynamic effects generated by the effective stroke and by the recovery stroke of the ciliary beat cycle.