Analytical and Numerical Approaches to Asymptotic Problems in Analysis

Unique book on Reaction-Advection-Diffusion problems

This dissertation aims at developing robust numerical methodologies to solve advective-diffusive-reactive systems that provide accurate and physical solutions for a wide range of input data (e.g., Péclet and Damköhler numbers) and for complicated geometries. It is well-known that physical quantities like concentration of chemical species and the absolute temperature naturally attain non-negative values. Moreover, the governing equations of an advective-diffusive-reactive system are either elliptic (in the case of steady-state response) or parabolic (in the case of transient response) partial differential equations, which possess important mathematical properties like comparison principles, maximum-minimum principles, non-negativity, and monotonicity of the solution. It is desirable and in many situations necessary for a predictive numerical solver to meet important physical constraints. For example, a negative value for the concentration in a numerical simulation of reactive-transport will result in an algorithmic failure. The objective of this dissertation is two fold. First, we show that many existing popular numerical formulations, open source scientific software packages, and commercial packages do not inherit or mimic fundamental properties of continuous advective-diffusive-reactive systems. For instance, the popular standard single-field Galerkin formulation produces negative values and spurious node-to-node oscillations for the primary variables in advection-dominated and reaction-dominated diffusion-type equations. Furthermore, the violation is not mere numerical noise and cannot be neglected. Second, we shall provide various numerical methodologies to overcome such difficulties. We critically evaluate their performance and computational cost for a wide range of Péclet and Damköhler numbers. We first derive necessary and sufficient conditions on the finite element matrices to satisfy discrete comparison principle, discrete maximum principle, and non-negative constraint. Based on these conditions, we obtain restrictions on the computational mesh and generate physics-compatible meshes that satisfy discrete properties using open source mesh generators. We then show that imposing restrictions on computational grids may not always be a viable approach to achieve physically meaningful non-negative solutions for complex geometries and highly anisotropic media. We therefore develop a novel structure-preserving numerical methodology for advective-diffusive reactive systems that satisfies local and global species balance, comparison principles, maximum principles, and the non-negative constraint on coarse computational grids. This methodology can handle complex geometries and highly anisotropic media. The proposed framework can be an ideal candidate for predictive simulations in groundwater modeling, reactive transport, environmental fluid mechanics, and modeling of degradation of materials. The framework can also be utilized to numerically obtain scaling laws for complicated problems with non-trivial initial and boundary conditions.

This special volume provides a broad overview and insight in the way numerical methods are being used to solve the wide variety of problems in the electronics industry. Furthermore its aim is to give researchers from other fields of application the opportunity to benefit from the results wich have been obtained in the electronics industry. * Complete survey of numerical methods used in the electronic industry* Each chapter is selfcontained* Presents state-of-the-art applications and methods* Internationally recognised authors

This new edition incorporates new developments in numerical methods for singularly perturbed differential equations, focusing on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics.

Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage