On enforcing maximum principles, comparison principles, monotone property, and the non-negative constraint for linear/nonlinear and steady-state/transient diffusion-type equations

The governing equations for reactive transport, under appropriate conditions, satisfy many important mathematical properties and physical constraints. To name a few: maximum principles, comparison principles, monotone property, and the non-negative constraint. Unfortunately, conventional numerical methods to solve the reactive transport equations violate these important mathematical properties and physical constraints. For example, one may obtain non-physical values (negative concentrations) under conventional numerical methods, particularly for cases with highly heterogeneous flow fields, irregular grids and full tensor dispersion. Negative concentrations cannot be tolerated when transport is coupled to nonlinear reactions.

In this presentation, we discuss the performance of finite element formulations with respect to maximum principles, comparison principles, monotone property, and the non-negative constraints. We shall show that the approach of placing restrictions on the mesh and time-step is not always possible. We shall also show that the consistent Newton-Raphson, which is a standard iterative procedure for solving nonlinear problems, needs preserve these important mathematical properties for semi-linear and quasi-linear diffusion models. We will show that the Pao’s method (which is based on Picard linearization) with mesh restrictions can be a viable option for solving semi-linear diffusion models. We shall also discuss the numerical convergence properties of the Pao method. Finally, we shall present optimization-based methods for steady state and transient diffusion models to meet some of these properties. The performance of various approaches will be illustrated using bimolecular (both slow and fast) reactions.