Difference scheme alternating direction

An alternating direction scheme. Further developments are concerned with the heat conduction equation of two independent variables that can serve as test vehicles for the difference schemes to be presented ... 

The scheme ascribed to Peaceman and Rachford provides some realization of this idea and refers to implicit alternating direction schemes. The present values y = and y = of this difference scheme are put... 

Thus, three different examples of interest bring out the indisputable merit of the alternating direction method and unveil its potential. From what has been said above it follows that the factorized schemes find some range of applications solely in rectangles and parallelepipeds and no more. The only exception is the case B = BiB, Ba = E + tRc, where Ri and R2 are triangle operators, by means of which it is possible to generate a lower-order approximation only under the condition r = 0 h" ). 

The alternating direction scheme ascribed to Duglas and Rachford and associated with the difference operator — Dx x comes second ... 

The scheme ascribed to Peaceman and Rachford provides some realization of this idea and refers to implicit alternating direction schemes. The present values y = yn and y = yn+1 of this difference scheme are put together with the intermediate value y = j/n+1 2, a formal treatment of which is the value of y at moment t = tn+]/2 = tn + r/ 2. The passage from the nth layer to the (n + 1)th layer can be done in two steps with the appropriate spacings 0.5 r ... 

The contaminant transport model, Eq. (28), was solved using the backwards in time alternating direction implicit (ADI) finite difference scheme subject to a zero dispersive flux boundary condition applied to all outer boundaries of the numerical domain with the exception of the NAPL-water interface where concentrations were kept constant at the 1,1,2-TCA solubility limit Cs. The ground-water model, Eq. (31), was solved using an implicit finite difference scheme subject to constant head boundaries on the left and right of the numerical domain, and no-flux boundary conditions for the top and bottom boundaries, corresponding to the confining layer and impermeable bedrock, respectively, as... 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

Perhaps the major bottleneck in modifier- and direct-input-adaptation schemes lies in the estimation of this gradient information. The finite-difference scheme used in the original ISOPE paper [19] is known to be inefficient for large-scale, slow and noisy processes. Hence, alternative techniques have been developed, which can be classified as either model-based approaches or perturbation-based approaches. 

Studies on solidification modeling have been largely directed towards macroscopic phenomena. A variety of numerical techniques have been used for such modeling studies. Among these are the finite difference method (FDM) with or without the alternate direction implicit (ADI) time-stepping scheme, the FEM, the boundary element method (BEM), the direct finite difference method (DFDM), and the control volume element (VFM) method. 

In principle, rac-lactide, a racemic mixmre of d- and L-lactide, may be polymerized in a stereoselective fashion. Depending on the stereoselection as the ROP proceeds, the resulting polymer may thus exhibit different stereoregularities these directly influence the thermal and mechanical properties of the produced PLAs. In this regard, isotactic PLA stereoblocks and PLA stereocomplexes, which are of interest for their thermal and mechanical properties, may be produced via the ROP of rac-lactide initiated by an achiral derivative, provided the polymerization proceeds via a chain-end stereocontrolled mechanism i.e., the last inserted lactide unit stereo-controls the insertion of the incoming monomer. This strategy has been first validated using salen-based aluminum complexes such as 16 (Scheme 16, top) to produce PLLA-PDLA isotactic stereoblocks [95, 96]. Alternatively, the chiral racemic salen aluminum complex 17 was found to be suitable for the parallel stereoselective synthesis of isotactic poly(D-lactide) and poly(L-lactide) from rac-... 

The reductive couphng of imines can follow different pathways, depending on the nature of the one-electron reducing agent (cathode, metal, low-valent metal salt), the presence of a protic or electrophihc reagent, and the experimental conditions (Scheme 2). Starting from the imine 7, the one-electron reduction is facihtated by the preliminary formation of the iminiiim ion 8 by protonation or reaction with an electrophile, e.g., trimethylsilyl (TMS) chloride. Alternatively, the radical anion 9 is first formed by direct reduction of the imine 7, followed by protonation or reaction with the electrophile, so giving the same intermediate a-amino radical 10. The 1,2-diamine 11 can be formed from the radical 10 by dimerization (and subsequent removal of the electrophile) or addition to the iminium ion 8, followed by one-electron reduction of the so formed aminyl radical. In certain cases/conditions the radical 9 can be further reduced to the carbanion 12, which then attacks the... 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

The insertion of unsaturated molecules into metal-carbon bonds is a critically important step in many transition-metal catalyzed organic transformations. The difference in insertion propensity of carbon-carbon and carbon-nitrogen multiple bonds can be attributed to the coordination characteristics of the respective molecules. The difficulty in achieving a to it isomerization may be the reason for the paucity of imine insertions. The synthesis of amides by the insertion of imines into palladium(II)-acyl bonds is the first direct observation of the insertion of imines into bonds between transition metals and carbon (see Scheme 7). The alternating copolymerization of imines with carbon monoxide (in which the insertion of the imine into palladium-acyl bonds would be the key step in the chain growth sequence), if successful, should constitute a new procedure for the synthesis of polypeptides (see Scheme 7).348... 

To synthesize new surfactants, having incorporated both structural elements, the known siloxanyl modified halogenated esters and ethers of dicyclopentadiene [5] were treated with different amines according to the reaction scheme. Triethylamine yielded quaternary ammonium salts directly. Alternatively, after reaction with diethylamine or morpholine, the isolated siloxanyl-modified tertiary amines were also converted to quaternary species. To obtain anionic surfactants, the halogenated precursors were initially reacted with n-propylamine. In subsequent reaction steps the secondary amines formed were converted with maleic anhydride into amides, and the remaining acid functions neutralized. Course and rate of each single reaction strongly depended on the structure of the initial ester or ether compound and the amine applied. The basicity of the latter played a less important role [6]. 

Very often, however, these dyads are not isolated in the solid state and interact with neighboring ones at least along one preferential direction. In this case, we can distinguish two important situations the alternated spin chain and the spin ladder. As shown in Scheme 5, the alternated spin chain is characterized with two different magnetic interactions, noted / and aJ with 0 < a < 1. Note that if a = 0, one recovers the singlet-triplet behavior while if a = 1, we are in the presence of a uniform spin chain. The spin ladder is also characterized by two J values, noted J// and J in the following. 

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