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WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms

R. Donat, M. C. Martí, P. Mulet

TL;DR

This work extends the conservative ED chromatography model to generalized Langmuir-type adsorption isotherms, proving that the map $\boldsymbol W(\boldsymbol c)$ is a global bijection with a computable inverse $\boldsymbol C(\boldsymbol w)$, enabling a well-posed hyperbolic-parabolic system. By exploiting the eigenstructure of the flux Jacobian, the authors develop a characteristic-based WENO scheme integrated with a second-order IMEX time integrator, yielding oscillation-free, high-resolution solutions even at sharp fronts. Theoretical results on the Jacobian structure and eigenvalues underpin the numerical approach, which is validated on experiments with Tóth's isotherms, showing superior suppression of spurious oscillations compared to component-wise methods and favorable performance for sharp interfaces. The framework promises robust chromatographic simulations with high accuracy and can be extended to SMB models and broader isotherm families in future work.

Abstract

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modelled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modelling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (Appl. Numer. Math. 123 (2018) 22-42) to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with Tóth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions.

WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms

TL;DR

This work extends the conservative ED chromatography model to generalized Langmuir-type adsorption isotherms, proving that the map is a global bijection with a computable inverse , enabling a well-posed hyperbolic-parabolic system. By exploiting the eigenstructure of the flux Jacobian, the authors develop a characteristic-based WENO scheme integrated with a second-order IMEX time integrator, yielding oscillation-free, high-resolution solutions even at sharp fronts. Theoretical results on the Jacobian structure and eigenvalues underpin the numerical approach, which is validated on experiments with Tóth's isotherms, showing superior suppression of spurious oscillations compared to component-wise methods and favorable performance for sharp interfaces. The framework promises robust chromatographic simulations with high accuracy and can be extended to SMB models and broader isotherm families in future work.

Abstract

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modelled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modelling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (Appl. Numer. Math. 123 (2018) 22-42) to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with Tóth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions.
Paper Structure (10 sections, 5 theorems, 53 equations, 11 figures, 1 table)

This paper contains 10 sections, 5 theorems, 53 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. The mathematical structure of the ED model.
  3. Characteristic-based WENO schemes
  4. IMEX schemes
  5. Numerical experiments
  6. Experiment 1
  7. Experiment 2
  8. Experiment 3
  9. Experiment 4
  10. Conclusions

Key Result

Theorem 2.1

Let $\mathcal{I}, \mathcal{J}, \mathcal{P}$ be real intervals and Assume that for any $p\in\mathcal{P}$ the function $\chi(\cdot, p)\colon \mathcal{I}^N\to\mathcal{J}^N$ is bijective and that for any $\boldsymbol w\in\mathcal{J}^N$ there exists a unique $p=p(\boldsymbol w)\in\mathcal{P}$ such that Then $\boldsymbol W$ is bijective and the function $\boldsymbol C\colon\mathcal{J}^N\to\mathcal{I}^

Figures (11)

  • Figure 1: Experiment 1. Numerical solutions obtained with MUSCL, COMP-UPW5, COMP-UPW1, COMP-GLF, CHR-GLF and CHR-UPW schemes for $\nu=1$, $D_a=0$ (a) and $D_a=10^{-5}$ (c) for $T=1, 4, 8$ and $11$. Enlarged views of approximate concentrations for components 1 and 2 at $T=8$ are given in (b) and (d).
  • Figure 2: Experiment 1. Numerical solutions obtained with MUSCL, COMP-UPW5, COMP-UPW1, COMP-GLF, CHR-GLF and CHR-UPW schemes with $D_a=0$ and $\nu=0.95$ (a), $\nu=0.9$ (c) and $\nu=0.6$ (e) for $T=1, 4, 8$ and $11$. Plots (b), (d) and (f) are enlarged views of (a), (c) and (e) respectively, for $T=8$.
  • Figure 3: Experiment 1. Numerical solutions obtained with MUSCL, COMP-UPW5, COMP-UPW1, COMP-GLF, CHR-GLF and CHR-UPW schemes with $D_a=10^{-5}$ and $\nu=0.95$ (a), $\nu=0.9$ (c) and $\nu=0.6$ (e) for $T=1, 4, 8$ and $11$. Plots (b), (d) and (f) are enlarged views of (a), (c) and (e) respectively, for $T=8$.
  • Figure 4: Experiment 1. Performance of MUSCL, COMP-UPW5, COMP-UPW1, COMP-GLF, CHR-GLF and CHR-UPW methods for $D_a=0$, $T=1$ (left) and $T=11$ (right) and values of $\nu=1$ (a) and (b), $\nu=0.9$ (c) and (d) and $\nu=0.6$ (e) and (f).
  • Figure 5: Experiment 1. Performance of MUSCL, COMP-UPW5, COMP-UPW1, COMP-GLF, CHR-GLF and CHR-UPW methods, with the $L^1$-error computed discarding the $2\%$ of the largest errors. We have used $D_a=0$, $T=11$ and values of $\nu=1$ (a), $\nu=0.9$ (b) and $\nu=0.6$ (c).
  • ...and 6 more figures

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Proposition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.2
  • proof