Many reaction mechanisms have been developed over the past few decades to predict flame characteristics. A detailed reaction mechanism can predict flame characteristics well, but at a high computational cost. The reason for reducing reaction mechanisms is to reduce the computational time needed to simulate a problem. The focus of this work is on the validity of reduced methane-air combustion mechanisms, particularly pertaining to satisfying the entropy inequality. While much of this work involves a two-step reaction mechanism developed by Dr. Charles Westbrook and Dr. Frederick Dryer, some consideration is given to the four-step and three-step mechanisms of Dr. Norbert Peters. These mechanisms are used to simulate the Flame A experiment from Sandia National Laboratories. The two-step mechanism of Westbrook and Dryer is found to generate results that violate the entropy inequality. Modifications are made to the two-step mechanism simulation in an effort to reduce these violations. Two new mechanisms, Mech 1 and Mech 2, are developed from the original two-step reaction mechanism by modifying the empirical data constants in the Arrhenius reaction form. The reaction exponents are set to the stoichiometric coefficients of the reaction, and the concentrations computed from a one-dimensional flame simulation are matched by changing the Arrhenius parameters. The new mechanisms match experimental data more closely than the original two-step mechanism and result in a significant reduction in entropy inequality violations. The solution from Mech 1 had only 9 cells that violated the entropy inequality, while the original two-step mechanism of Westbrook and Dryer had 22,016 cells that violated the entropy inequality. The solution from Mech 2 did not have entropy inequality violations. The method used herein for developing the new mechanisms can be applied to more complex reaction mechanisms.

Symmetrical Multilevel Diversity Coding (SMDC) is a network compression problem for which a simple separate coding strategy known as superposition coding is optimal in terms of achieving the entire admissible rate region. Carefully constructed induction argument along with the classical subset entropy inequality of Han played a key role in proving the optimality. This thesis considers a generalization of SMDC for which, in addition to the randomly accessible encoders, there is also an all-access encoder. It is shown that superposition coding remains optimal in terms of achieving the entire admissible rate region of the problem. Key to our proof is to identify the supporting hyperplanes that define the boundary of the admissible rate region and then build on a generalization of Han's subset inequality. As a special case, the (R0,Rs) admissible rate region, which captures all possible tradeoffs between the encoding rate, R0, of the all-access encoder and the sum encoding rate, Rs, of the randomly accessible encoders, is explicitly characterized. To provide explicit proof of the optimality of superposition coding in this case, a new sliding-window subset entropy inequality is introduced and is shown to directly imply the classical subset entropy inequality of Han.

During the last half century there has been a resurgence of interest in Monge's 18th century mass transportation problem, with most of the activity limited to continuous spaces. This thesis, consequently, develops techniques based on mass transportation for the purpose of obtaining tight concentration inequalities in a discrete setting. Such inequalities on n-fold products of graphs, equipped with product measures, have been well investigated using combinatorial and probabilistic techniques, the most notable being martingale techniques. The emphasis here, is instead on the analytic viewpoint, with the precise contribution being as follows. We prove that the modified log-Sobolev inequality implies the transportation inequality in the first systematic comparison of the modified log-Sobolev inequality, the Poincar inequality, the transportation inequality, and a new variance transportation inequality. The duality shown by Bobkov and Gtze of the transportation inequality and a generating function inequality is then utilized in finding the asymptotically correct value of the subgaussian constant of a cycle, regardless of the parity of the length of the cycle. This result tensorizes to give a tight concentration inequality on the discrete torus. It is interesting in light of the fact that the corresponding vertex isoperimetric problem has remained open in the case of the odd torus for a number of years. We also show that the class of bounded degree expander graphs provides an answer, in the affirmative, to the question of whether there exists an infinite family of graphs for which the spread constant and the subgaussian constant differ by an order of magnitude. Finally, a candidate notion of a discrete Ricci curvature for finite Markov chains is given in terms of the time decay of the Wasserstein distance of the chain to its stationarity. It can be interpreted as a notion arising naturally from a standard coupling of Markov chains. Because of its natural definition, ease of calculation, and tensoring property, we conclude that it deserves further investigation and development. Overall, the thesis demonstrates the utility of using the mass transportation problem in discrete isoperimetric and functional inequalities.

Les phénomènes de transport en milieux poreux sont étudiés depuis près de deux siècles, cependant les travaux concernant les milieux fortement poreux sont encore relativement peu nombreux. Les modèles couramment utilisés pour les poreux classiques (lits de grains par exemple) sont peu applicables pour les milieux fortement poreux (les mousses par exemple), un certain nombre d’études ont été entreprises pour combler ce manque. Néanmoins, les résultats expérimentaux et numériques caractérisant les pertes de charge dans les mousses sont fortement dispersés. Du fait des progrès de l’imagerie 3D, une tendance émergente est la détermination des paramètres des lois d’écoulement à partir de simulations directes sur des géométries reconstruites. Nous présentons ici l’utilisation d’une nouvelle approche cinétique pour résoudre localement les équations de Navier-Stokes et déterminer les propriétés d’écoulement (perméabilité, dispersion, ...). / A novel kinetic scheme satisfying an entropy condition is developed, tested and implemented for the simulation of practical problems. The construction of this new entropic scheme is presented. A classical hyperbolic system is approximated by a discrete velocity vector kinetic scheme (with the simplified BGK collisional operator), but applied to an inviscid compressible gas dynamics system with a small Mach number parameter, according to the approach of Carfora and Natalini (2008). The numerical viscosity is controlled, and tends to the physical viscosity of the Navier-Stokes system. The proposed numerical scheme is analyzed and formulated as an explicit finite volume flux vector splitting (FVS) scheme that is very easy to implement. It is close in spirit to Lattice Boltzmann schemes, but it has the advantage to satisfy a discrete entropy inequality under a CFL condition and a subcharacteristic stability condition involving a cell Reynolds number. The new scheme is proved to be second-order accurate in space. We show the efficiency of the method in terms of accuracy and robustness on a variety of classical benchmark tests. Some physical problems have been studied in order to show the usefulness of both schemes. The LB code was successfully used to determine the longitudinal dispersion of metallic foams, with the use of a novel indicator. The entropic code was used to determine the permeability tensor of various porous media, from the Fontainebleau sandstone (low porosity) to a redwood tree sample (high porosity). These results are pretty accurate. Finally, the entropic framework is applied to the advection-diffusion equation as a passive scalar.