Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one‐dimensional second‐order wave equation

In this article, we analyze a residual‐based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one‐dimensional second‐order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L² error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862–901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L²‐norm under mesh refinement. The order of convergence is proved to be , when p‐degree piecewise polynomials with are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and superconvergent solutions. Our computational results show higher convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L²‐norm converge to unity at rate while numerically they exhibit and rates, respectively. Numerical experiments are shown to validate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461–1491, 2015     

Molecular dynamics comparative study of methane–nitrogen and methane–nitrogen–ethane systems

This work concerns the site–site interaction study of 256 particles using the Buckingham potential model. We have calculated the new parameters of the Buckingham potential using an iterative algorithm with a mean square method. This adapted model allows determining the characteristics for each state point. We have applied this model to study the liquefied natural gas LNG properties for methane-nitrogen and methane–nitrogen–ethane mixtures by molecular dynamics. We have calculated the thermodynamic, dynamic and structural properties for both the microcanonical NVT and the isothermal-isobaric NPT ensembles of binary and ternary systems from the SP₁ to SP₉ points. Then, we have compared the results between binary and ternary systems. We have obtained a good prediction on transport properties. From the calculated values of self-diffusion coefficient and viscosity, we have confirmed the liquid state of the liquefied natural gas LNG system.     

A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u ^″ = f (t, u), which arise in a wide variety of engineering applications. We prove the L ² stability of the LDG scheme and optimal L ² error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p +?1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p +?1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p +?3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p +?2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥?1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.     

An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems

Numerical Algorithms - In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear...     

Study of femtosecond soliton dynamics in photonic crystal fiber using the moment method

We investigate the dynamics of femtosecond solitons in photonic crystal fibers (PCFs) by including high-order dispersion terms until to sixth-order in the generalized nonlinear Schrödinger equation, in addition to the nonlinear effects of the self phase modulation, self steepening and Raman scattering. We calculate theoretically the pulse parameters using the moment method. In the case of the fundamental soliton, our computed equations are coupled and difficult to solve analytically. However, we use the finite difference method to calculate numerically pulse parameters using an initially hyperbolic secant pulse at 1550-nm with different peak powers along 10m-PCF. Our numerical results show that the nonlinear regimes allow obtaining pulse compressions and initial pulse amplitudes. Furthermore, we remark a pulse broadening, and weak shifts of the peak power positions and frequencies in the critical and dispersive regimes. The use of an initial chirp provides a better pulse compressions and especially for low input powers. Also, the initial positive chirp reduces the optimal compression position lengths, while the negative one increases them. Therefore, we conclude that our theoretical calculations and numerical simulation results show that the moment method associated with the finite differences method is effective for the study of femtosecond pulse dynamics in PCFs.     

Statistical binary decay of35Cl+24Mg at ≈ 8 MeV/nucleon

High spin rotational bands in ¹⁶² Lu and ¹⁶⁴ Lu have been studied through the ¹³⁹ La( ²⁸ Si,5n) and ¹³⁹ La( ²⁹⁽³⁰⁾ Si,4(5)n) reactions respectively. For both nuclei the yrast sequence which is associated with the πh _11/2 ?νi _13/2 configuration shows the signature inversion feature.     

Study of the fusion-fission process in the 35Cl +24Mg reaction

Fusion-fission and fully energy-damped binary processes of the ³⁵Cl+²⁴Mg reaction were investigated using particle-particle coincidence techniques at a ³⁵Cl bombarding energy of E_lab≈ 8 MeV/nucleon. Inclusive data were also taken in order to determine the partial wave distribution of the fusion process. The fragment-fragment correlation data show that the majority of events arises from a binary-decay process with a relatively large multiplicity of secondary light-charged particles emitted by the two primary excited fragments in the exit channel. No evidence is observed for ternary-breakup processes, as expected from the systematics recently established for incident energies below 15 MeV/nucleon and for a large number of reactions. The binary-process results are compared with predictions of statistical-model calculations. The calculations were performed using the Hauser-Feshbach method, based on the available phase space at the scission point of the compound nucleus. This new method uses temperature-dependent level densities and its predictions are in good agreement with the presented experimental data, thus consistent with the fusion-fission origin of the binary fully-damped yields. Received: 12 March 1998     

Green synthesis and characterization of silver nanoparticles using Juniperus communis leaf extract: Catalytic activity in real-outdoor conditions and electrochemical properties

The present study investigates the green synthesis of stable silver nanoparticles using Juniperus communis leaf aqueous extract at room temperature. Synthesized silver nanoparticles (AgNPs) were characterized with different techniques such as UV–vis spectroscopy, Fourier transforms infrared (FT-IR) spectroscopy, X-ray diffraction (XRD), Scanning electron microscopy (SEM-EDAX) and electrochemical method. Photocatalytic and anti-bacterial activities of synthesized AgNPs are evaluated based on the obtained result showed an efficient inhibition growth for gram negative P. Aeruginosa, E. Coli, and gram positive bacteria S.aureus. The AgNPs exhibited an excellent photocatalytic activity toward the degradation of methylene blue both indoor and outdoor, under sunlight, an efficiency of 95% was achieved. As an easy and environmentally friendly process, AgNPs based on Juniperus communis leaf extract could be applied for the degradation of pollutants and wastewater treatment.