Gaussian random fields with Matérn covariance functions are popular models in spatial statistics and machine learning. In this work, we develop a spatio-temporal extension of the Gaussian Matérn fields formulated as solutions to a stochastic partial differential equation. The spatially stationary subset of the models have marginal spatial Matérn covariances, and the model also extends to Whittle-Matérn fields on curved manifolds, and to more general non-stationary fields. In addition to the parameters of the spatial dependence (variance, smoothness, and practical correlation range) it additionally has parameters controlling the practical correlation range in time, the smoothness in time, and the type of non-separability of the spatio-temporal covariance. Through the separability parameter, the model also allows for separable covariance functions. We provide a sparse representation based on a finite element approximation, that is well suited for statistical inference and which is implemented in the R-INLA software. The flexibility of the model is illustrated in an application to spatio-temporal modeling of global temperature data.

The multivariate Gaussian distribution is widely used in many statistical applications due to its appealing features. However, real-world data often violate its as-sumptions, showing skewness and/or tail-thickness. One can use other parametric distributions which incorporate skewness and kurtosis in the model such as the skew-normal distribution proposed by Azzalini (ASN) and its various extensions, which are widely popular among practitioners. In this thesis, the first part addresses the singularity problem of the Fisher information matrix in the ASN family when skewness parameters are set to zero by introducing the multivariate modified skew-normal (MSN) distribution. Moreover, defining the MSN with an alternative parameterization, we can make the profile log-likelihood of its skewness parameter bounded, unlike the ASN family. While the MSN distribution resolves these two problems of the ASN distribution, it sacrifices stochastic features like closed-form expressions for marginal and conditional distributions, as well as for moments, making it less practical for statistical applications. In the second part, we tackle one limitation of the ASN and the MSN distributions, namely their incapability of handling heavy-tailed data. We introduce the skew-normal-Tukey-h distribution by applying Tukey’s h transformation to the marginals of an ASN-distributed random vector but with the alternative parameterization used in defining the MSN distribution, enabling it to capture heavy-tail features. Varying Tukey’s h parameters for different marginals can have different tail-thickness, unlike the skew-t distribution, another ASN extension for modelling heavy tails. The skew-normal-Tukey-h distribution preserves appealing stochastic properties of the skew-normal distribution, making it apt for many statistical applications. Non-Gaussian models can also be constructed using copulas. In the third part, we introduce a copula-based model for non-Gaussian, non-stationary replicated spatial data. We study its extremal properties and inferential procedures. Yet another approach for modelling non-Gaussianity is transforming a Gaussian random vector, like the Tukey g-and-h random field, defined by applying the Tukey g-and-h transformation to induce skewness and tail-thickness. In the fourth part, we provide a parameter estimation method for the Tukey g-and-h random field using high-performance computing (HPC), for large-scale geostatistical applications. Additionally, we provide the tile-low-rank approximation of the covariance matrix to facilitate even larger applications.

An OpenFOAM® based hybrid-central solver called reactingPimpleCentralFoam is validated to compute hydrogen-based detonations. This solver is a pressure-based semi-implicit compressible flow solver based on central-upwind schemes of Kurganov and Tadmor. This solver possesses the features of standard OpenFOAM® solvers namely, rhoCentralFoam, reactingFoam and pimpleFoam. The solver utilizes Kurganov & Tadmor schemes for flux splitting to solve the high-speed compressible regimes with/without hydrodynamic discontinuity. In this work, we present the validation results that were obtained from one-dimensional (1D) and two-dimensional (2D) simulations with detailed chemistry. We consider three different mixtures that fall into the categories of weakly unstable mixture (2H2 + O2 + 3.76Ar and 2H2 + O2 + 10Ar), and moderately unstable mixture (2H2 + O2 + 3.76N2), based on their approximate effective activation energy. We performed the numerical simulations in both laboratory frame of reference (LFR) and shock-attached frame of reference (SFR) for the 1D cases. The 1D simulation results obtained using this solver agree well with the steady-state calculations of Zel’dovich von Neumann Döring (ZND) simulations with an average error below 1% in all cases. For the 2D simulations, circular hot-spots were used to perturb the initially-planar detonations to develop into spatio-temporally unstable detonation front. The convergence is declared when the front does not deviate much from the CJ speed (Chapman-Jouguet) and the regularity of cellular pattern on the numerical smoke foils reaches a steady state. We have verified from our preliminary studies that the SFR-based simulations are computationally cheaper in comparison to the LFR simulations and that the required grid resolution is always lesser in the former than the latter to reach the same level of accuracy (in terms of speed of the detonation front and cell sizes from the numerical smoke foil). We have also verified that at least 24 points per induction zone length (for weakly unstable mixture) and 40 points per induction zone length (for moderately unstable mixture) are required to sufficiently resolve the detonation structures that are independent of grids, boundary and initial conditions. Further reduction in computational cost of approximately 50% is achieved by using non-uniform grids, which converge effectively to the same solutions in comparison to the results from twice the number of grids with uniform resolution.

The detonation front’s unstable structure leads to an unsteady and threedimensional (3D) phenomenon that renders the study of the cell cycle challenging. Traditionally, fundamental studies are carried out in narrow channels where the detonation behaviour is very peculiar (quasi two-dimensional with velocity deficit). In this study, we propose a fully experimental approach to study the cell cycle in the case of non-marginal detonations. The cell cycle is characterized through three techniques: systematic and statistical analysis of soot foil, planar laser-induced fluorescence on nitric oxide, and Rayleigh scattering. These techniques provide measurements for cell size, induction length, and local shock speed, respectively. The work is carried out in the 2H2-O2-3.76Ar mixture at 20 kPa and 293 K. These conditions ensure that the cell pattern is extremely regular, thus, a shot-to-shot reconstruction of the cell cycle is possible. The cell widths follow a normal distribution, from which a quantitative parameter (2σ/λ) is proposed to assess the cell regularity, experimentally. The evolution of the speed and of the local induction length are reconstructed along the cell cycle. The results agree with the available data for narrow channels and constitute the first of their kind for 3D detonation (i.e., non-marginal detonation). Two methods are proposed and compared to determine the Zel’dovich–von Neumann–Döring (ZND) induction length Δi from the presented experimental measurements. The technique can be applied to mixtures where the mean cell width is a meaningful parameter from highly regular to irregular mixtures.