A full analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the unsteady one dimensional heat equation

The present work is an extension of our previous works ,  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for parabolic equations. We aim in this work (and some forthcoming studies) at getting higher order (both in time and space) finite volume approximations for the exact solution of parabolic equations using the class of spatial generic meshes introduced recently in [13]. We focus in the present contribution on the one dimensional heat equation and its implicit finite volume scheme described in [3]. The implicit finite volume scheme approximating the one dimensional heat equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal. The finite volume approximate solution is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allows us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using the same linear systems involved in the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximations of the second, third, and fourth spatial derivatives of the exact solution. The computation of the approximation of these high-order derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [12] without any restrictive condition on the spatial mesh. A full analysis for the stated theoretical results as well as some numerical examples supporting the theory is presented. The results obtained in the present study are based essentially on two facts. The first fact is the use of the results provided in [3] which state the convergence order of the finite volume approximate solution in several norms. The second fact is the comparison between the stated new higher order approximations and suitable auxiliary finite volume approximations.     

A theoretical analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the one dimensional wave equation

This paper details our note [6] and it is an extension of our previous works  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for second order hyperbolic equations with Dirichlet boundary conditions on general nonconforming multidimensional spatial meshes introduced recently in [14]. We aim in this work (and some forthcoming studies) to get higher order (both in time and space) finite volume approximations for the exact solution of hyperbolic equations using the class of spatial generic meshes introduced recently in [14] on low order schemes from which the matrices used to compute the discrete solutions are sparse. We focus in the present contribution on the one dimensional wave equation and on one of its implicit finite volume schemes described in [4]. The implicit finite volume scheme approximating the one dimensional wave equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal, symmetric and definite positive. The finite volume approximate solution of the basic finite volume scheme is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allow us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using linear systems whose matrices are the same ones used to compute the discrete solution of the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximation of some high order spatial derivatives of the exact solution. The computation of the approximation of these high order spatial derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [13] without any condition. To reach the above results, a theoretical framework is developed and some numerical examples supporting the theory are presented. Some of the tools of this framework are new and interesting and they are stated in the one space dimension but they can be extended to several space dimensions. In particular a new and useful a prior estimate for a suitable discrete problem is developed and proved. The proof of this a prior estimate result is based essentially on the decomposition of the solution of the discrete problem into the solutions of two suitable discrete problems. A new technique is used in order to get a convenient finite volume approximation whose discrete time derivatives of order up to order two are also converging towards the solution of the wave equation and their corresponding time derivatives.     

Synthese Originale de 5-Aryl (ou 5-benzyl)-2[(1-Diethoxyphosphonyl)methyl]-1,3,4-oxadiazoles par Action du Phosphonomethylhydrazide sur les Imidates N-Acyles

The phosphonomethylhydrazide 2a reacts with N-acylated imidates 3a–d to give the corresponding 5-aryl (or 5-benzyl)-2-[(1-diethoxyphosphonyl)methyl]-1,3,4-oxadiazoles 4a–d after the elimination of ethanamide 5. Compounds 2a–e are prepared by the action of triethyl phosphonoacetate 1 with hydrazine and its derivatives. The structures of 1,3,4-oxadiazoles 4a–d and hydrazides 2a–e have been unequivocally confirmed by means of IR, ¹ H, ¹³ C, ³¹ P NMR and mass spectrometry.     

The Stereoselective Synthesis of Conjugated Allylsilanes

2-trimethylsilylethylidenetriphenylphosphorane (5) (Seyferth–Wittig reagent) reacts stereoselectively with the carbonyl compounds 6a–f to give the conjugated allylsilanes 7a–f, each as a mixture of E-and Z-isomers. The stereoselectivity of reactions of E-cinnamaldehyde (6c) with 5 has been investigated at different temperatures. A successful E-stereoselective synthesis of 7c was achieved by reacting 5 with E-cinnamaldehyde (6c) under the conditions of a Wittig–Schlosser modification reaction. Structures of the allylsilanes 7a–f were deduced by compatible analytical and spectroscopic (IR, ¹H NMR, ¹³C NMR, and GC/MS) measurements. An assignment of the E:Z ratios of 7a–f is based on their ¹H NMR spectral data.     

Organophosphorus Chemistry,Part 37: Synthesis and Insecticidal Activities of Some Phosphonate Adducts Derived from 6-(Aryliminomethyl)-furobenzopyran-5-ones

Dialkyl phosphites attack 6-(Aryliminomethyl)furobenzopyran-5-ones at the imine-carbon atom to give new phosphonate 1:1 adducts. The obtained compounds regenerate the starting materials upon thermolysis under reduced pressure. Structures of the new products were attested by compatible analytical and spectroscopic measurements. Insecticidal activity tests assured that some new compounds show marked potency against adults of Aphis gossypii (Glover), which infests cotton crops.

Supplemental materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfur, and Silicon and the Related Elements to view the free supplemental file.     

Preparation,Thermolysis, and Photolysis of Some New Thiophosphoramidates Derived from 3-Methyl-2-Benzothiazolinone Hydrazone

3-Methyl-2-benzothiazolinone hydrazone (1) reacts with dialkyl phosphorothiochloridates 2a,b in the presence of a base, to give the respective dialkylthiophosphorylated hydrazones 3a,b. Upon thermolysis, compound 3b yields bi(3-methylbenzothiazole-2-iminyl) (4). Exposure of 3b to sunlight in methanol results in the formation of 3-methyl-2-benzothiazolinone (5). When the same experiment was carried on the starting hydrazone 1, bis(3-methyl- benzothiazole-2-iminyl)diazine (6) was formed. Elemental analyses and spectroscopic details are presented for the new compounds.

Supplemental materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfur, and Silicon and the Related Elements to view the free supplemental file.     

Synthese de 5-Aryl-3-[(1-dialcoxyphosphonyl)methyl]-1,2,4-triazoles a Partir d'Imidates N-Chloroacyles

A variety of 5-aryl-3-[(1-dialcoxyphosphonyl)methyl]-1,2,4-triazoles 3 has been synthesized efficiently by treatment of trialkylphosphites with 5-aryl-3-chlorométhyl-1,2,4-triazoles 2 . Compound 2 has been prepared by action of N-chloroacylimidates 1 with hydrazines. The structure of triazoles 2 and 3 have been unequivocally confirmed by means of IR, 1 H, 13 C, 31 P NMR spectroscopy and mass spectra.     

STUDIES ON ORGANOPHOSPHORUS COMPOUNDS VI1–5: Utility of Lawesson's Reagent for Synthesis of Thiaphosphorine,Thioxanthene, and Thiaphosphole Derivatives

Arylidene malonate derivatives 2a–c reacted with Lawesson's reagent (1) LR in equimolar ratio to yield the oxathiaphosphorine derivatives 3a–c. The behaviour of LR towards cyclic ketones was also examined and yielded the thioxanthene derivatives 5a,b. On the other hand, arylidene pyrazolone 8 reacted with LR to give the phosphole 10. Aminobenzenethiophene 11 reacted with LR under reflux to produce the corresponding thiazaphosphorine 12.     

A Novel and Convenient Preparation of 1-(γ-Aminoalkyl)-Substituted 1,2,4-Triazoles

1-(α-Aminomethyl)-1,2,4-triazoles readily add to enol ethers or enamides to give novel classes of compounds, 1-(γ-amino-α-alkoxy)propyl- (8) and 1-(γ-amino-α-amido)propyl- (10) substituted 1,2,4-triazoles, in excellent yields.     

EDXRF,FTIR, and XRD characterization of low calcium oxalate urinary stones collected from arid area

Low calcium oxalate urinary stones from the kidney, bladder, and ureter have been collected from the arid area (Taif, Saudi Arabia). After careful washing and drying of the collected stones, the samples were converted into a contamination-free homogenous fine powder with a particles' size smaller than 50 μm. The processed urinary stone powders were studied using attenuated total reflection Fourier transform infrared spectroscopy (ATR-FTIR), laboratory setup and synchrotron radiation X-ray diffraction (XRD), and energy-dispersive X-ray fluorescence (EDXRF). The activated function groups, quantitative phase analysis, and the semi-quantitative elemental analysis of the present urinary stones were identified. Seventeen elements were measured in most of the urinary stone samples. The significant elements are Ca, P, S, Cl, Zn, K, Fe, and Cu, whereas other elements were found alternatively in a few number of urinary stone samples. It was recognized that Ca exists with low concentration, which indicates the presence of different calcium phases even with low percentages. In 33% of the urinary stones, the phosphorus (P) was not measured, but there were high concentrations of sulfur (S) and low concentrations of Ca up to 2.15 ± 0.05%. The ATR-FTIR results indicate that the most compounds of the present urinary stones were urea and cystine combined with low ratios of calcium oxalate and calcium phosphate compounds. The quantitative phase analysis of the XRD of selected samples proves the presence of the cystine, urea, and calcium oxalate phases with different weight percent.