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A High Accuracy Symplectic Scheme for Advection Diffusion Reaction Models in Bioseparation

Farjana Siddiqua, Catalin Trenchea

TL;DR

This paper addresses advection–diffusion–reaction transport with nonlinear adsorption in chromatography membranes. It develops a time-stepping strategy based on a symplectic implicit midpoint method and a finite element spatial discretization to achieve second-order accuracy and good mass conservation. The authors prove stability and error estimates for constant, affine, and nonlinear adsorption cases, including existence of fully discrete solutions and a time-integrated formulation for nonlinear adsorption, with numerical tests validating the theory. The results enhance the reliability and predictive capability of chromatography models, facilitating design and optimization of bioseparation membranes.

Abstract

We analyze an advection-diffusion-reaction problem with non-homogeneous boundary conditions that models the chromatography process, a vital stage in bioseparation. We prove stability and error estimates for both constant and affine adsorption, using the symplectic one-step implicit midpoint method for time discretization and finite elements for spatial discretization. In addition, we perform the stability analysis for the nonlinear, explicit adsorption in the continuous and semi-discrete cases. For the nonlinear, explicit adsorption, we also complete the error analysis for the semi-discrete case and prove the existence of a solution for the fully discrete case. The numerical tests validate our theoretical results.

A High Accuracy Symplectic Scheme for Advection Diffusion Reaction Models in Bioseparation

TL;DR

This paper addresses advection–diffusion–reaction transport with nonlinear adsorption in chromatography membranes. It develops a time-stepping strategy based on a symplectic implicit midpoint method and a finite element spatial discretization to achieve second-order accuracy and good mass conservation. The authors prove stability and error estimates for constant, affine, and nonlinear adsorption cases, including existence of fully discrete solutions and a time-integrated formulation for nonlinear adsorption, with numerical tests validating the theory. The results enhance the reliability and predictive capability of chromatography models, facilitating design and optimization of bioseparation membranes.

Abstract

We analyze an advection-diffusion-reaction problem with non-homogeneous boundary conditions that models the chromatography process, a vital stage in bioseparation. We prove stability and error estimates for both constant and affine adsorption, using the symplectic one-step implicit midpoint method for time discretization and finite elements for spatial discretization. In addition, we perform the stability analysis for the nonlinear, explicit adsorption in the continuous and semi-discrete cases. For the nonlinear, explicit adsorption, we also complete the error analysis for the semi-discrete case and prove the existence of a solution for the fully discrete case. The numerical tests validate our theoretical results.

Paper Structure

This paper contains 19 sections, 18 theorems, 109 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Notation and Preliminaries
  4. Assumptions and preliminary results
  5. Variational Formulation
  6. Semi-Discrete in Space Approximation
  7. Fully-discrete approximation
  8. Time-integrated finite element formulation
  9. Time-Dependent Analysis
  10. Construction of $\hat{C}$
  11. Nonlinear, Explicit Isotherm
  12. Semi-discrete in space error estimate
  13. Existence of the solution to the fully discrete system
  14. Stability of the solution to the fully discrete system
  15. Affine Isotherm
  16. ...and 4 more sections

Key Result

Lemma 1

For all $v\in H_{0,\Gamma_{\text{in}}}^1(\Omega)$, there exists a constant ${\Tilde{K}}_{PF}$ such that

Figures (5)

  • Figure 1: Domain $\Omega$
  • Figure 2: Langmuir Isotherm: Temporal rate of convergence of BE and Midpoint, $T=1.0$, $h=1/128$. Notice that Midpoint is giving order 2 whereas BE is giving order 1.
  • Figure 3: Langmuir Isotherm: Comparison of total mass for exact solution, BE, Midpoint, $T=1.0$, $h=1/128$, $dt=1/8$. Notice that BE overestimates total mass rather than underestimates.
  • Figure 4: Langmuir isotherm: Plot of concentration while using BE (Left) $\&$ Midpoint (Right), $\Omega=[0,2]\times [0,10]$, $f=0$, $g=1$, $T=3.0$, $h=1/128$, $dt=1/128$, ${\bf u} =(0,2x(x-2))$, $D=I$.
  • Figure 5: Langmuir isotherm: Comparison of total mass, $\Omega=[0,2]\times [0,10]$, $f=0$, $g=1$, $T=3.0$, $h=1/128$, $dt=1/128$, ${\bf u} =(0,2x(x-2))$, $D=I$.

Theorems & Definitions (39)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Lemma 4
  • ...and 29 more