a b s t r a c tIn [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.

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We present theoretical results for the dipole moment, linear polarizability, and first hyperpolarizability of the urea and thiourea molecules in solid phase. The in-crystal electric properties were determined by applying a supermolecule approach in combination with an iterative electrostatic scheme, in which the surrounding molecules are represented by point charges. It is found for both urea and thiourea molecules that the influence of the polarization effects is mild for the linear polarizability, but it is marked for the dipole moment and first hyperpolarizability. The replacement of oxygen atoms by sulfur atoms increases, in general, the electric responses. Our second-order Møller-Plesset perturbation theory based iterative scheme predicts for the in-crystal dipole moment of urea and thiourea the values of 7.54 and 9.19 D which are, respectively, increased by 61% and 58%, in comparison with the corresponding isolated values. The result for urea is in agreement with the available experimental result of 6.56 D. In addition, we present an estimate of macroscopic quantities considering explicit unit cells of urea and thiourea crystals including environment polarization effects. These supermolecule calculations take into account partially the exchange and dispersion effects. The results illustrate the role played by the electrostatic interactions on the static second-order nonlinear susceptibility of the urea crystal.

The dipole moment (μ), linear polarizability (α), and first hyperpolarizability (β(tot)) of the asymmetric unit of L-arginine phosphate (LAP) monohydrate crystal are investigated using the supermolecule approach in combination with an iterative electrostatic polarization scheme. Environment polarization effects are attained by assuring the convergence of the dipole moment of LAP embedded in the polarization field of the surrounding molecules whose atomic sites are treated as point charges. The results obtained show that in the presence of the embedding charges, the value of μ is increased by 9% but the static values of α and β(tot) are decreased, respectively, by 3% and 13%, as compared with the isolated situation. The MP2/6-311+G(d) model predicts for the in-crystal dipole moment the converged value of 33 D, in good concordance with the available experimental result of 32 D. Our estimates for the converged results of α and β(tot) are, respectively, 22.51×10(-24) and 5.01×10(-30) esu. Dispersion effects are found to have a small impact on the nonlinear optical responses of LAP in the visible region. In addition, MP2/6-311G results obtained for β(tot) by using isolated and embedded LAP dimers show that crystal packing effects have a significant contribution of the electrostatic interactions. Our results suggest that the role of the crystal environment is to minimize the effects of the intermolecular interactions in the electric properties. That is, μ and β(tot) gain a more additive character in the presence of the field of the embedding charges. This is specially marked for β(tot).

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