Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented. (C) 2003 Elsevier Science Ltd. All rights reserved.

A mechanistic model for carbon dioxide corrosion of mild steel in the presence of protective iron carbonate films - Part 3: Film growth model

A model of iron carbonate (FeCO3) film growth is proposed, which is an extension of the recent mechanistic model of carbon dioxide (CO2) corrosion by Nesic, et al. In the present model, the film growth occurs by precipitation of iron carbonate once saturation is exceeded. The kinetics of precipitation is dependent on temperature and local species concentrations that are calculated by solving the coupled species transport equations. Precipitation tends to build up a layer of FeCO3 on the surface of the steel and reduce the corrosion rate. On the other hand, the corrosion process induces voids under the precipitated film, thus increasing the porosity and leading to a higher corrosion rate. Depending on the environmental parameters such as temperature, pH, CO2 partial pressure, velocity, etc., the balance of the two processes can lead to a variety of outcomes. Very protective films and low corrosion rates are predicted at high pH, temperature, CO2 partial pressure, and Fe2+ ion concentration due to formation of dense protective films as expected. The model has been successfully calibrated against limited experimental data. Parametric testing of the model has been done to gain insight into the effect of various environmental parameters on iron carbonate film formation. The trends shown in the predictions agreed well with the general understanding of the CO2 corrosion process in the presence of iron carbonate films. The present model confirms that the concept of scaling tendency is a good tool for predicting the likelihood of protective iron carbonate film formation.

A partial velocity approach to subcycling structural dynamics

Subcycling, or the use of different timesteps at different nodes, can be an effective way of improving the computational efficiency of explicit transient dynamic structural solutions. The method that has been most widely adopted uses a nodal partition. extending the central difference method, in which small timestep updates are performed interpolating on the displacement at neighbouring large timestep nodes. This approach leads to narrow bands of unstable timesteps or statistical stability. It also can be in error due to lack of momentum conservation on the timestep interface. The author has previously proposed energy conserving algorithms that avoid the first problem of statistical stability. However, these sacrifice accuracy to achieve stability. An approach to conserve momentum on an element interface by adding partial velocities is considered here. Applied to extend the central difference method. this approach is simple. and has accuracy advantages. The method can be programmed by summing impulses of internal forces, evaluated using local element timesteps, in order to predict a velocity change at a node. However, it is still only statistically stable, so an adaptive timestep size is needed to monitor accuracy and to be adjusted if necessary. By replacing the central difference method with the explicit generalized alpha method. it is possible to gain stability by dissipating the high frequency response that leads to stability problems. However. coding the algorithm is less elegant, as the response depends on previous partial accelerations. Extension to implicit integration, is shown to be impractical due to the neglect of remote effects of internal forces acting across a timestep interface. (C) 2002 Elsevier Science B.V. All rights reserved.

Fast, scalable master equation solution algorithms. IV. Lanczos iteration with diffusion approximation preconditioned iterative inversion

In this paper we propose a second linearly scalable method for solving large master equations arising in the context of gas-phase reactive systems. The new method is based on the well-known shift-invert Lanczos iteration using the GMRES iteration preconditioned using the diffusion approximation to the master equation to provide the inverse of the master equation matrix. In this way we avoid the cubic scaling of traditional master equation solution methods while maintaining the speed of a partial spectral decomposition. The method is tested using a master equation modeling the formation of propargyl from the reaction of singlet methylene with acetylene, proceeding through long-lived isomerizing intermediates. (C) 2003 American Institute of Physics.

Selecting methods to solve multi-well master equations

There are several competing methods commonly used to solve energy grained master equations describing gas-phase reactive systems. When it comes to selecting an appropriate method for any particular problem, there is little guidance in the literature. In this paper we directly compare several variants of spectral and numerical integration methods from the point of view of computer time required to calculate the solution and the range of temperature and pressure conditions under which the methods are successful. The test case used in the comparison is an important reaction in combustion chemistry and incorporates reversible and irreversible bimolecular reaction steps as well as isomerizations between multiple unimolecular species. While the numerical integration of the ODE with a stiff ODE integrator is not the fastest method overall, it is the fastest method applicable to all conditions.

Synthesis and characterization of mono- and bis-ligand zinc(II) and cadmium(II) complexes of the di-2-pyridylketone Schiff base of S-benzyl dithiocarbazate (Hdpksbz) and the X-ray crystal structures of the [Zn(dpksbz)2] and [Cd(dpksbz)NCS]2 complexes

New mono- and bis-chelated zinc(II) and cadmium(II) complexes of formula, [M(dpksbz)NCS] (dpksbz = anionic form of the di-2-pyridylketone Schiff base of S-benzyldithiocarbazate) and [M(dpksbz)(2)] (M = Zn-II, Cd-II) have been prepared and characterized. The structure of the bis-ligand complex, [Zn(dpksbZ)(2)] has been determined by X-ray diffraction. The complex has a distorted octahedral geometry in which the ligands are coordinated to the zinc(II) ion as uninegatively charged tridentate chelates via the thiolate sulfur atoms, the azomethine nitrogen atoms and the pyridine nitrogen atoms. The distortion from a regular octahedral geometry is attributed to the restricted bite angles of the Schiff base ligands. X-ray structural analysis shows that the [Cd(dpksbz)NCS](2) complex is a centrosymmetric dimer in which each of the cadmium(II) ions adopts a five-coordinate, approximately square-pyramidal configuration with the Schiff base acting as a tetradentate chelating agent coordinating a cadmium(II) ion via one of the pyridine nitrogen atoms, the azomethine nitrogen atom and the thiolate sulfur atom; the second pyridine nitrogen atom is coordinated to the other cadmium(II) ion of the dimer. The fifth coordination position around each cadmium(II) is occupied by an N-bonded thiocyanate ligand. (C) 2003 Elsevier Science Ltd. All rights reserved.

An approach to verifying and debugging simulation models governed by ordinary differential equations: Part 1. Methodology for residual generation

For dynamic simulations to be credible, verification of the computer code must be an integral part of the modelling process. This two-part paper describes a novel approach to verification through program testing and debugging. In Part 1, a methodology is presented for detecting and isolating coding errors using back-to-back testing. Residuals are generated by comparing the output of two independent implementations, in response to identical inputs. The key feature of the methodology is that a specially modified observer is created using one of the implementations, so as to impose an error-dependent structure on these residuals. Each error can be associated with a fixed and known subspace, permitting errors to be isolated to specific equations in the code. It is shown that the geometric properties extend to multiple errors in either one of the two implementations. Copyright (C) 2003 John Wiley Sons, Ltd.

An approach to verifying and debugging simulation models governed by ordinary differential equations: Part 2. Residuals analysis and a case study

In Part 1 of this paper a methodology for back-to-back testing of simulation software was described. Residuals with error-dependent geometric properties were generated. A set of potential coding errors was enumerated, along with a corresponding set of feature matrices, which describe the geometric properties imposed on the residuals by each of the errors. In this part of the paper, an algorithm is developed to isolate the coding errors present by analysing the residuals. A set of errors is isolated when the subspace spanned by their combined feature matrices corresponds to that of the residuals. Individual feature matrices are compared to the residuals and classified as 'definite', 'possible' or 'impossible'. The status of 'possible' errors is resolved using a dynamic subset testing algorithm. To demonstrate and validate the testing methodology presented in Part 1 and the isolation algorithm presented in Part 2, a case study is presented using a model for biological wastewater treatment. Both single and simultaneous errors that are deliberately introduced into the simulation code are correctly detected and isolated. Copyright (C) 2003 John Wiley Sons, Ltd.

Cytotoxic iron chelators: Characterization of the structure, solution chemistry and redox activity of ligands and iron complexes of the di-2-pyridyl ketone isonicotinoyl hydrazone (HPKIH) analogues

Di-2-pyridyl ketone isonicotinoyl hydrazone (HPKIH) and a range of its analogues comprise a series of monobasic acids that are capable of binding iron (Fe) as tridentate (N,N,O) ligands. Recently, we have shown that these chelators are highly cytotoxic, but show selective activity against cancer cells. Particularly interesting was the fact that cytotoxicity of the HPKIH analogues is maintained even after complexation with Fe. To understand the potent anti-tumor activity of these compounds, we have fully characterized their chemical properties. This included examination of the solution chemistry and X-ray crystal structures of both the ligands and Fe complexes from this class and the ability of these complexes to mediate redox reactions. Potentiometric titrations demonstrated that all chelators are present predominantly in their charge-neutral form at physiological pH (7.4), allowing access across biological membranes. Keto-enol tautomerism of the ligands was identified, with the tautomers exhibiting distinctly different protonation constants. Interestingly, the chelators form low-spin (diamagnetic) divalent Fe complexes in solution. The chelators form distorted octahedral complexes with Fe-II, with two tridentate ligands arranged in a meridional fashion. Electrochemistry of the Fe complexes in both aqueous and non-aqueous solutions revealed that the complexes are oxidized to their ferric form at relatively high potentials, but this oxidation is coupled to a rapid reaction with water to form a hydrated (carbinolamine) derivative, leading to irreversible electrochemistry. The Fe complexes of the HPKIH analogues caused marked DNA degradation in the presence of hydrogen peroxide. This observation confirms that Fe complexes from the HPKIH series mediate Fenton chemistry and do not repel DNA. Collectively, studies on the solution chemistry and structure of these HPKIH analogues indicate that they can bind cellular Fe and enhance its redox activity, resulting in oxidative damage to vital biomolecules.

A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework. (C) 2002 Elsevier Science B.V. All rights reserved.