We present a series of new tunneling models based on a reaction class approach. Reaction class consists of all reactions that have the same reactive moiety. One can expect that reactions in the same class share similarities in the shape of the potential energy surfaces along the reaction path. By exploring such similarities, we propose to use reaction path information from the parent (smallest) reaction in calculations of tunneling contributions of larger reactions in the class. This significantly reduces the computational cost while maintaining the accuracy of the model.

We investigate the use of the reaction-class approach within the integrated molecular + molecular orbital (IMOMO) methodology for improving energetic information of chemical reactions. We have tested this approach using two classes of hydrogen abstraction reactions. One is abstraction from saturated hydrocarbons and the other from unsaturated hydrocarbons. For saturated hydrocarbon systems, this approach yields average unsign errors of the order of 1 kcal/mol in the reaction energy and about 0.2 kcal/mol in the barrier height. The errors are larger in the unsaturated hydrocarbon systems and are of the order of 2 kcal/mol. Analysis of the performance shows that this approach provides a practical and cost-effective tool for studying reactions involving large molecules.

The methods of thermogravimetric analysis (TGA) and differential scanning calorimettry (DSC) have been used to study the thermal decomposition of ammonium perchlorate (AP). TGA curves obtained under both isothermal and nonisothermal conditions show a characteristic slowdown at the extents of conversion, α = 0.30-0.35. DSC demonstrates that in this region the process changes from an exothermic to an endothermic regime. The latter is ascribed to dissociative sublimation of AP. A new computational technique (advanced isoconversional method) has been used to determine the dependence of the effective activation energy (Eα) on α for isothermal and nonisothermal TGA data. At α > 0.1, the Eα dependencies obtained from isothermal and nonisothermal data are similar. By the completion of decomposition (α → 1) the activation energy for the isothermal and nonisothermal decomposition respectively rises to ∼110 and ∼130 kJ mol^-1, which are assigned to the activation energy of sublimation. The initial decomposition (α → 0) shows the activation energy of 90 kJ mol^-1 for the isothermal decomposition and 130 kJ mol^-1 for the nonisothermal decomposition. The difference is explained by different rate-limiting steps, which are nucleation and nuclei growth for isothermal and nonisothermal decompositions, respectively.

A procedure is outlined for adapting Krylov subspace methods to solving approximately the underdetermined linear systems that arise in path following (continuation, homotopy) methods. This procedure, in addition to preserving the usual desirable features of Krylov subspace methods, has the advantages of satisfying orthogonality constraints exactly and of not introducing ill-conditioning through poor scaling.

GMRES(k) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications.

Richards' equation (RE) is often used to model flow in unsaturated porous media. This model captures physical effects, such as sharp fronts in fluid pressures and saturations, which are present in more complex models of multiphase flow. The numerical solution of RE is difficult not only because of these physical effects but also because of the mathematical problems that arise in dealing with the nonlinearities. The method of lines has been shown to be very effective for solving RE in one space dimension. When solving RE in two space dimensions, direct methods for solving the linearized problem for the Newton step are impractical. In this work, we show how the method of lines and Newton-iterative methods, which solve linear equations with iterative methods, can be applied to RE in two space dimensions. We present theoretical results on convergence and use that theory to design an adaptive method for computation of the linear tolerance. Numerical results show the method to be effective and robust compared with an existing approach.

This review covers both the history and present state of the kinetics of thermally stimulated reactions in solids. The traditional methodology of kinetic analysis, which is based on fitting data to reaction models, dates back to the very first isothermal studies. The model fitting approach suffers from an inability to determine the reaction model uniquely,and this does not allow reliable mechanistic conclusions to be drawn even from isothermal data. In non-isothermal kinetics, the use of the traditional methodology results in highly uncertain values of Arrhenius parameters that cannot be compared meaningfully with isothermal values. An alternative model-free methodology is based on the isoconversional method. The use of this model-free approach in both isothermal and non-isothermal kinetics helps to avoid the problems that originate from the ambiguous evaluation of the reaction model. The model-free methodology allows the dependence of the activation energy on the extent of conversion to be determined. This, in turn, permits reliable reaction rate predictions to be made and mechanistic conclusions to be drawn.