On Enforcing Maximum Principles and Element-Wise Species Balance for Advective-Diffusive-Reactive Systems

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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.

Advection-diffusion-reaction equations, Non-self-adjoint operators, Comparison principles, Maximum principle, Non-negative constraint, Monotone property, Monotonicity, Local and global species balance, Oscillatory chemical reactions, Chaotic mixing, Mesh restrictions, Anisotropic M-uniform meshes, Angle conditions, Least-squares mixed formulations, Convex optimization, Pao's method, Picard's method, Newton-Raphson methods