A stabilization technique for coupled convection‐diffusion‐reaction equations

Abstract

Partial differential equations having diffusive, convective, and reactive terms appear in the modeling of a large variety of processes in several branches of science. Often, several species or components interact with each other, rendering strongly coupled systems of convection-diffusion-reaction equations. Exact solutions are available in extremely few cases lacking practical interest due to the simplifications made to render such equations amenable by analytical tools. Then, numerical approximation remains the best strategy for solving these problems. The properties of these systems of equations, particularly the lack of sufficient physical diffusion, cause most traditional numerical methods to fail, with the appearance of violent and nonphysical oscillations, even for the single equation case. For systems of equations, the situation is even harder due to the lack of fundamental principles guiding numerical discretization. Therefore, strategies must be developed in order to obtain physically meaningful and numerically stable approximations. Such stabilization techniques have been extensively developed for the single equation case in contrast to the multiple equations case. This paper presents a perturbation-based stabilization technique for coupled systems of one-dimensional convection-diffusion-reaction equations. Its characteristics are discussed, providing evidence of its versatility and effectiveness through a thorough assessment. KEYWORDS convection-diffusion equation, differential equations, finite element methods, stability Int J Numer Methods Eng. 2018;116:43-65.wileyonlinelibrary.com/journal/nme

A multifield coupled thermo‐chemo‐mechanical theory for the reaction‐diffusion modeling in photovoltaics

“…The initial concentrations of the chemical species over the whole volume are prescribed as C init = (0.64, 0.32, 0, 0, 0) T with the unit of mol/m 3 . To avoid possible numerical instability, 52 the mesh size of the finite element model is set to 3 mm, and the total number of elements for the EVA layer is 2860. The contour plots of moisture diffusion in the 85 • C/85% RH damp heat test obtained from simulation are shown in Figure 4 and compared with the corresponding experimental EL images at different time points.…”

“…The initial concentrations of the chemical species over the whole volume are prescribed as $$$ {\mathbf{C}}_{\mathrm{init}}={\left(0.64,0.32,0,0,0\right)}^{\mathrm{T}} $$$ with the unit of $$$ \mathrm{mol}/{\mathrm{m}}^3 $$$. To avoid possible numerical instability, 52 the mesh size of the finite element model is set to 3 mm, and the total number of elements for the EVA layer is 2860.…”

Stabilization of coupled convection–diffusion-reaction equations for continuum dislocation transport

“…where the c k constants solely depend on the boundary conditions given by (34), and λ k and v (k) are the 2m eigenvalues and eigenvectors of a 2m × 2m matrix constructed with the transport matrices defined in (16) as follows…”

“…Taking the single equation case as the departure point, a stabilization technique based on coefficient perturbations was proposed and proven to be effective for systems of linear one-dimensional convectiondiffusion-reaction equations in steady state arising from different branches of science and engineering [34]. The main objective of this communication is to extend and to apply this stabilization technique for dislocation transport problems expressed in terms of the total and geometrically necessary dislocation densities, including its non-linearity in the transient regime.…”

“…Subsequently, a stabilization technique for a single steady state and linear convection-diffusion-reaction equation, originally proposed in [13], is reviewed. The extension of the stabilization technique extension to systems of coupled convection-diffusion-reaction equations, as proposed in [34], is discussed at the end of this section. Section 4 assesses the effectiveness of the stabilization technique for the system of dislocation transport equations through two numerical examples comparing stabilized results with those obtained with the classical Bubnov-Galerkin scheme.…”