Efficient and Reliable Mathematical Modeling Techniques for Multi-Phase Environmental Flows

This research is on using recent techniques of Software Quality Assurance (SQA) and developing Verification and Validation and Uncertainty Quantification (VVUQ) tools to improve mathematical models of contaminant, sediment, and air-bubble transport in natural aquatic bodies. A comprehensive toolkit for VVUQ is developed for: a) code and calculation verification with Methods of Exact Solutions (MES), Method of Manufactured Solutions (MMS), and cross-code verification; b) Richardson extrapolation for code and calculation verification; c) model validation via common statistical methods and model skill assessment metrics; d) quantification of uncertainty in numerical discretization of PDEs. In the next section of this dissertation, seven new closed-form analytical solutions of scalar transport equation devised for code verification with MES. The set of developed analytical solutions was complete in the sense that it is able to check nonlinearity as well as spatial and temporal non-homogeneity in all terms of scalar ADR equation. In addition, a 2D analytical description of air-bubbles distribution in hydraulic jumps was derived. The new analytical model was validated versus various empirical datasets via common model skill assessment metrics. The experimental dataset was also used to design an analytical-empirical model for air entrainment/detrainment in the two-phase flows of hydraulic jumps. In the rest of this dissertation the emphasis was changed from two-phase flows of air and water into two-phase flows of sediment particles and water. First, a comprehensive assessment of former methods of computing total sediment discharge with Einstein's method was conducted. Sequential and parallel subroutines of computing the Einstein's integrals with existing methods developed. Then local and global accuracy, convergence behavior, singularities, CPU time and parallelization efficiency were studied via common metrics of model skill assessment. Second, four new methods of computing Einstein's integrals for calculation of total sediment discharge were devised: a) a numerical technique which exploits the similarity of integrand functions to devise a numerical recycling of values for reduction of computational time; b) nested adaptive Gauss-Kronrod quadrature; c) perturbation techniques to find a fast asymptotic series representation to approximate the Einstein integrals; d) semi-analytical solutions based on Gauss hypergeometric function. All of the developed methods were benchmarked against machine-precision-accurate results. Efficiency of those new methods in parallel computing was evaluated.