1 Introduction

Chromatographic separation, a fundamental technique for the isolation and purification of substances, utilizes solid stationary and fluid mobile phases. Elution chromatography, specifically, involves the injection of a mixture into an HPLC column to separate its components based on their interactions with the stationary phase. Introduced nearly six decades ago [1], gradient elution has revolutionized analytical liquid chromatography (LC) by continuously varying the eluent concentration, allowing for the separation of all components in a single run. This is achieved by adjusting the mobile phase composition, often through the addition of a modulator. Despite its extensive use in analytical LC, gradient elution has been less common in preparative LC [2, 3]. However, significant advancements have been made by Snyder et al. [4,5,6,7,8], who have focused on the development of gradient elution techniques for preparative LC, aiming to optimize the separation system. Their work has been supported by both theoretical studies and practical experiments, which have demonstrated the efficiency of gradient elution under ideal conditions for both theoretical and experimental systems.

The efficiency of gradient elution in preparative liquid chromatography was quickly demonstrated in theoretical studies [9, 10] and ideal operating conditions were found for theoretical [11, 12] and experimental separation systems [13]. A discussion on the prediction of retention times in gradient elution mode using isocratic experimental data is given in [14]. The authors in the article [15] discusses the experimental design and re-parameterization of the Neue–Kuss model to enhance the accuracy and precision of predicting isocratic retention factors from gradient measurements in reversed-phase liquid chromatography. The study referenced in [16] showcases the Tecan Freedom EVO, a high-throughput purification tool that enables continuous gradient elution in RoboColumn experiments by delivering liquid more consistently. Jurgen et al. [17] presents a systematic comparison of the elution behavior of plasmid DNA across three widely-used anion exchange resins, employing both linear gradient and isocratic elution experiments. Gradient elution is now commonly used for separating mixtures of components having a broad range of retentivity [18,19,20,21]. The method is commonly used in the separation of biopolymers to take advantage of the increased peak capacity that comes with it [22, 23].

Since the beginning of research in this area, there has been a need for an efficient modeling approach for in-depth studies of the technique. This results in highly linear conductivity gradients. Most current gradient elution models are developed for linear concentration ranges. An association between modulator concentrations and component-specific adsorption equilibrium constants is crucial for studying migration processes in the presence of solvent gradients. Until now, linear gradients have been commonly used because they can be defined by simple theoretical relationships [24, 25]. In reversed-phase systems, the LSS model [8, 26, 27] is the most familiar gradient efficient model.

Various difficulties are encountered in the simulation of gradient elution chromatographic systems. Particularly when dealing with nonlinear convection-dominated partial differential equations (PDEs) of chromatographic models incorporating Langmuir isotherms and high column efficiency. In such scenarios, sharp fronts and narrow peaks can appear in the solutions in a finite time [28, 29]. Because of non-availability of analytical solutions to such model equations, appropriate numerical methods must be applied to ensure stability, efficiency, and physically realistic solutions. A number of numerical techniques are available in the literature, such as Non-Oscillatory Finite Difference methods like total variance diminishing, Essentially Non-Oscillatory (ENO), and Weighted Essentially Non-Oscillatory (WENO) schemes. In contrast, the finite element approaches are capable of dealing with complex boundary conditions and/or geometries. Conventional finite element methods, such as standard Galerkin methods, are unstable and inaccurate for such models and can produce nonphysical solutions. Hughes et al. have proposed the streamline diffusion (SD) process, which subsequently eliminates oscillations based on the numerical solution [30,31,32,33,34,35,36]. Nonetheless, hyperbolic PDEs are known to be amenable to explicit solutions, the SD process failed due to its implicit nature in time [37, 38]. Therefore, the discontinuous Galerkin (DG) method was later on introduced to resolve these concerns and to generate stable and high order accurate solutions [33,34,35,36]. Reed and Hill proposed this method for solving hyperbolic PDEs [39]. After that, various DG methods for non-linear hyperbolic problems have been formulated. Lesaint and Raviart published the first mathematical study of the method in 1974 [40], demonstrating its applicability to the neutron transport framework. Hulme [41, 42] has proposed an analogous strategy for ODEs just a year before Reed and Hill devised the DG approach, with the exception of the solution approximation being continuous rather than discontinuous. Bassi and Rebay [38] have made significant progress in the study of compressible Navier–Stokes equations using DG space discretization and a new approach for choosing numerical-fluxes. Cockburn and his co-authors have investigated DG methods more deeply in their various articles [43,44,45,46]. To prevent numerical oscillations, the DG methods utilizes the ideas of slope limiters and numerical fluxes. As these methods are highly parallelizable, they are well suited for complex boundary conditions and complicated geometries.

Numerous differential models exist that characterize the migration of sample zones along chromatographic columns [2]. These models vary in their levels of complexity. The general rate model, acknowledged for its complexity, encompasses all phenomena occurring in the mobile phase, within the pores, and on the stationary phase surface, offering detailed insights into chromatographic processes. In contrast, the simplest ideal model overlooks kinetic processes that cause band broadening, focusing solely on the impact of thermodynamic processes on band profile evolution. Meanwhile, the equilibrium-dispersive (ED) model strikes a balance, proving effective for optimizing chromatographic operations. In most cases of practical interest, the ED model lacks a closed-form analytical solution, necessitating the use of numerical solutions. Numerous methods exist to compute numerical solutions for the ED Model. The researchers in [47] discusses the development of an orthogonal collocation method that utilizes moving finite elements to simulate fixed-bed adsorbers. This method is based on the Equilibrium Theory for wave transitions and the shock layer theory for shock transitions. A modified approach to the Craig scheme for obtaining a numerical solution of the ED model in gradient chromatography is presented in [48]. Another contribution discussed in [49] is the development and application of the Martin–Synge algorithm for computing the solution of the ED model.

In the present manuscript, the Runge–Kutta discontinuous Galerkin (RKDG) approach is developed to solve the equilibrium dispersive model considering gradient elution. The DG scheme utilizes finite element space discretization in the axial coordinate, which transforms the given PDEs into a system of ODEs. The resultant system is solved by employing a non-linearly stable explicit high order Runge–Kutta (RK) method that satisfies the total variation bounded property for ensuring the scheme’s positivity. The numerically simulated solutions are used to analyze the impact of gradient starting and ending times, solvent strength, and gradient slope on the profiles of concentration. The obtained results will be useful for understanding and upgrading gradient elution chromatography.

The present article is structured as follows: Sect. 2 provides a discussion on the Equilibrium Dispersive Model (EDM). Section 3 is dedicated to the derivation of the RKDG method within the EDM framework. Numerical case studies are examined in Sect. 4. Finally, Sect. 5 presents the conclusions.

2 The EDM for gradient elution

This section aims to introduce the Equilibrium Dispersive Model (EDM) for gradient elution. In the context of gradient elution chromatography, the dynamic behavior of elution profiles is analyzed utilizing the multi-component EDM. This model presupposes that the phase system remains in equilibrium. It is posited that the volume fraction of the non-retained solvent impacts the axial dispersion, Henry’s constants, and the coefficients of nonlinearity. Based on these assumptions, the mass-balance equations for a mixture of \(N_c\) species percolating through the packed-bed column are presented as follows:

$$\begin{aligned}&\frac{\partial c_i}{\partial t}+F\frac{\partial q_{i}^{*}(\textbf{c},\phi (t,z))}{\partial t} +u\frac{\partial c_i}{\partial z}\nonumber \\&=\frac{\partial }{\partial z}\left(\left(D_{z,i}(\phi (t,z))\frac{\partial c_i}{\partial z}\right)\right), \quad i=1,2,\ldots ,N_{c}. \end{aligned}$$
(1)

In Eq. 1, \(D_{z,i}\) represents the axial dispersion coefficient for the i-th component, \(c_i\) denotes the liquid phase concentration of the i-th component within the mixture, \(\textbf{c}\) is a vector representing the concentrations of all components, u is the bulk interstitial linear velocity, and \(N_c\) signifies the total number of components in the mixture. The variable t indicates time, while z refers to the distance along the axial coordinates. Additionally, \(q_i^*\) represents the concentration of non-equilibrium adsorbents for the i-th constituent. The symbol \(\phi\) indicates the volume fraction of the modified non-retained solvent in gradient elution. Moreover, F is defined as \(F = \left( \frac{1-\epsilon }{\epsilon }\right)\), where \(\epsilon\) denotes the total porosity of the system.

The separation of components in a mixture and their migration rates are influenced by their distribution between the mobile and solid phases. In gradient elution, the concentration of a component in the stationary phase, \(q_i^*\), is influenced by the concentration of the mobile phase modulator, its own local concentration, and the concentrations of all other components. Consequently, the isotherm, which delineates the equilibrium relationship between the concentrations of the i-th component in the liquid and solid phases, must account for the interdependence of the modulator and the components in sorption behavior. The simplest and most commonly applied model for describing this relationship in a mixture of \(N_c\) components is the Langmuir isotherm, which is formulated as follows:

$$\begin{aligned} q_i^{*}(\textbf{c},\phi )=\frac{K_{{H,i}(\phi )}c_i}{1+\sum \limits _{k=1}^{N_\textrm{c}}b_k(\phi )c_k}, \quad i=1,2,\ldots ,N_{c}, \end{aligned}$$
(2)

where

$$\begin{aligned} K_{ H,i}(\phi )=k_{ Hr,i}e^{-\alpha \phi }, \quad b_{i}(\phi )=b_{i}^{ref}e^{-\alpha \phi }, \quad D_{ z,i}(\phi )=D_{ zr,i}e^{-\gamma \phi }. \end{aligned}$$
(3)

The i-th component reference Henry’s constant is denoted by \(k_{Hr,i}\). In addition, \(b_{i}^{ref}\) specifies the degree of nonlinearity inherent in the adsorption isotherm for the i-th component of the mixture. Here, we aim to obtain the numerical solution for this model in both, linear (\(b_i^{ref} = 0\)) and nonlinear (\(b_i^{ref} \ne 0\)) forms. Additionally, \(\alpha\) and \(\gamma\) represent the specific solvent strengths, while \(\phi\) denotes the volume-percent of the modifying non-retained solvent.

For a column which is empty initially, the initial conditions (ICs) are expressed as:

$$\begin{aligned} c_i(0,z)=0, \quad q_{i}(0,z)=0,\quad \textrm{i}=1,2,3,\ldots ,N_{c}. \end{aligned}$$
(4)

Furthermore, rectangular injections are taken into account in this research. Thus, the boundary conditions (BCs) are stated as:

$$\begin{aligned} c_i(t,0)&=\left\{ \begin{array}{ll} c_{ i,\mathrm inj} ,\quad \text {if}\,\,\, 0\le t\le t_\textrm{inj}\,,\\ 0\,,\quad \quad t>t_\textrm{inj}\,. \end{array}\right. \,\quad {i}=1,2,3,\ldots ,N_{c}. \end{aligned}$$
(5)

These inlet BCs at the inlet are connected with Neumann BCs at the right end of column

$$\begin{aligned} \frac{\partial c_i}{\partial z}(t,L)=0,\quad i=1,2,\ldots ,N_\textrm{c}, \end{aligned}$$
(6)

where, the concentration of i-th component concentration in the sample injection is denoted by the symbol \(c_{i,\mathrm inj}\) and the injection time is represented by \(t_\textrm{inj}\).

The local migration speed of the solute is influenced by the concentration of the mobile phase modulator, represented as \(\phi (t,z)\), at any given point z and time t. The modulator is presumed to be unretained, and band broadening does not alter its profile within the column. As a result, an ideal model can be employed to describe the evolution of \(\phi (t,z)\). For a column with a length of L, the preferred model that accounts for changes in the modulator concentration in the liquid phase is outlined as follows:

$$\begin{aligned} \frac{\partial \phi (t,z)}{\partial t}+ u\frac{\partial \phi (t,z)}{\partial z} =0, \end{aligned}$$
(7)

with initial and boundary conditions

$$\begin{aligned} \phi (0,z)&=\phi _0,\quad 0\le z\le L,\end{aligned}$$
(8)
$$\begin{aligned} \phi (t,0)&=\left\{ \begin{array}{ll} \phi _0\,,\quad \quad \quad \quad \text {if}\quad \,\,\,t< t_{s},\\ \Phi (t-t_s) \,,\quad \text {if}\quad t_{s}\le t\le t_\textrm{e},\\ \phi _e\,,\quad \quad \quad \quad \,\,\text {if}\quad t>t_\textrm{e}, \end{array}\right. \end{aligned}$$
(9)

where, the starting and ending time are denoted by \(t_s\) and \(t_e\), respectively. Moreover, \(\phi _0\) and \(\phi _e\) represent the initial and final concentrations of the mobile phase modulator, respectively, and \(\Phi\) denotes the gradient profile that has been implemented. In the case of a linear gradient, the concentration of the modifier at the location z down the column depends on time and gradient slope which is given as:

$$\begin{aligned} \Phi (t)=\phi _0+ t\beta , \quad \text {and} \quad \beta =\frac{\phi _e-\phi _0}{t_e-t_s}, \end{aligned}$$
(10)

where \(\beta\) denotes the slope of the gradient. Elution will be isocratic if \(\beta =0\), and will be a negative gradient if \(\phi _0>\phi _e\). This article piques our interest in examining the numerical effects of these limited cases as well.

3 Derivation of discontinuous Galerkin scheme

In this section, we introduce, formulate, and implement a discontinuous Galerkin (DG) scheme to efficiently solve the nonlinear equilibrium dispersive model used in gradient elution chromatography. The model equations exhibit profound nonlinearity due to the nonlinear adsorption isotherms. The proposed DG scheme is designed to accurately capture the sharp discontinuities of chromatographic fronts with a minimal number of grid points, thereby reducing computational costs. This scheme offers second-order accuracy both in time and space. It employs a Runge–Kutta DG method along the axial coordinate, transforming the partial differential equations (PDEs) into a system of ordinary differential equations (ODEs). These ODEs are then solved using the Total Variation Bounding (TVB) Runge–Kutta method, which is an explicit nonlinear approach. The essential steps for integrating this numerical algorithm are outlined as follows:

  1. 1)

    Across the column’s axial-coordinate, a finite number of grid points divide the computing domain.

  2. 2)

    The suggested DG method is applied in axial-coordinate to generate a system of time dependent coupled ODEs.

  3. 3)

    The resulting system of ordinary differential equations (ODEs) is approximated using the Total Variation Bounding (TVB) Runge–Kutta method, which is nonlinear and explicit.

  4. 4)

    The algorithm is implemented using the C programming language and MATLAB software.

3.1 Implementation of DG-scheme

To implement the discretization of the Discontinuous Galerkin Finite Element (DG-FE) scheme, Eq. 1 is reformulated into two first-order equations. This modification is achieved by rewriting Eq. 1 as follows:

$$\begin{aligned} \frac{\partial }{\partial {t}}\left( c_{i}+F q (c_{i}^{*})\right) +\frac{\partial }{\partial {z}}\left( u c_{ i}-D_{z,i}(\phi )\frac{\partial c_i}{\partial z}\right) =0, \quad i=1,2,\ldots ,N_{c}, \end{aligned}$$
(11)

and defining the following functions

$$\begin{aligned} w=w(c_{i}):=c_{i}+Fq(c_{i}^{*}),\quad g=g(c_{i}):=\sqrt{D_{z,i}(\phi )}\frac{\partial c_{i}}{\partial {z}}, \end{aligned}$$
(12)

such that

$$\begin{aligned} f(c_{i},g(c_{i}))= u c_{i}-\sqrt{D_{z,i}(\phi )}\left[ \sqrt{D_{z,i}(\phi )}\frac{\partial c_{i}}{\partial {z}}\right] , \end{aligned}$$
(13)
$$\begin{aligned} f(c_{i},g(c_{i}))= u c_{i}-\sqrt{D_{z,i}(\phi )}g(c_{i}). \end{aligned}$$
(14)

By substituting Eqs. (12)–(14) in Eq. (11), The following PDE system is obtained:

$$\begin{aligned} \frac{\partial {w(c_{i})}}{\partial {t}}+ \frac{\partial { f}}{\partial {z}}(c_{i},g(c_{i}))=0, \end{aligned}$$
(15)
$$\begin{aligned} g(c_{i})=\sqrt{D_{z,i}(\phi )}\frac{\partial c_{i}}{\partial {z}}. \end{aligned}$$
(16)

Before using the proposed numerical technique, the spatial domain along the axial coordinate must be discretized. For simplicity, we will refer to the number of nodes as N, the discretization index as m, and the considered computational domain as [0, L]. \(\Delta z\) and \(z_m\) stand for the width of the mesh intervals and the centres of the cells, respectively, the coordinates \(z_{m-\frac{1}{2}}\) and \(z_{m+\frac{1}{2}}\) are the left and right boundaries of the interval \(\Omega _\textrm{m}\equiv \left[ z_\mathrm{m-\frac{1}{2}},z_\mathrm{m+\frac{1}{2}} \right]\) for \(1\le \textrm{m}\le {N}\), and \(\Omega =U\Omega _m\) union of partition of the whole domain.

Let \(w_k(t,z)\) be the approximate numerical solution of w(t, z) and \(g_k(t,z)\) be the approximate numerical solution of g(t, z), such that \(w_k(t,z)\) corresponds to finite-dimensional space for each \(t\in [0,t_{max}]\). The local solution within each element \(\Omega _m\) is approximated using a polynomial of order N. We define the approximation space as consisting of piecewise smooth polynomials of order \(N = N_p - 1\), as follows:

$$\begin{aligned} U_k=\{{{\mu \in L^m (\Omega ):\mu |_{\Omega m} \in \textit{P}^{p}(\Omega _m),\quad \forall \,\,\Omega _m \in \Omega ,\quad m=1,2,3,4...,N}}\}. \end{aligned}$$
(17)

Where \(\textit{P}^{p}(\Omega _m)\) refers to the set of p-th degree polynomials specified in \(\Omega _m\). It is worth noting that in \(U_k\), the functions may have jump discontinuity at the cell interface \(z_\mathrm{m+\frac{1}{2}}\). A weak formulation is required to obtain the approximate solution \(w_k(t,z)\). To obtain weak formulation, multiply Eqs. (15) and (16) by an arbitrary smooth test function \(\mu (z)\), then integrate by parts over \({\Omega _m}\), [28] we get

$$\int _{\Omega _m}\mu (z)\frac{\partial w (t,z) }{\partial t}dz=-\left( \mu \Big(z_{m+\frac{1}{2}}\Big)f\Big(c_{m+\frac{1}{2}},g_{m+\frac{1}{2}}\Big)- \mu \Big(z_{m-\frac{1}{2}}\Big) f\Big(c_{m-\frac{1}{2}},g_{m-\frac{1}{2}}\Big)\right) + \int _{\Omega _m}f(c,g)\frac{\partial u(z)}{\partial z}dz,$$
(18)
$$\begin{aligned} \int _{\Omega _m}\mu (z)g(c)dz=\mu (z)\sqrt{D_{z,i}(\phi )} c(t,z)|_{\Omega _m}-\int _{\Omega _m}\frac{\partial u(z)}{\partial z}\sqrt{D_{z,i}(\phi )}c(t,z)dz. \end{aligned}$$
(19)

Select \(\textit{P}_\ell (z)\) Legendre polynomials of order \(\ell\) as the local basis function to execute Eq. (17). \(L^{2}\)-orthogonality property of Legendre polynomial can be used in this procedure. Generally, orthogonal properties of Legendre polynomials are

$$\begin{aligned} \int _{-1}^{1}{} \textit{P}_q(x)\textit{P}_r(x)dx=\left\{ \begin{array}{ll} 0 \,\,\,, \quad \quad q\ne r,\\ \frac{2}{2r+1}, \quad q=r, \end{array}\right. \quad \text {or} \quad \int _{-1}^{1}(\textit{P}_q(x))^{2}dx=\frac{2}{2r+1}, \quad \textrm{for}\,\, q=r. \end{aligned}$$
(20)

For each \(z\in \Omega _m\) the numerical solution can be written as:

$$\begin{aligned} w_k(t,z)=\sum _{\ell =0}^{p}{w_{m}^{\ell }} \varphi _{\ell }(z), \quad g_k\left( c_k(t,z)\right) =\sum _{\ell =0}^{p}{g_{m}^{\ell }} \varphi _{\ell }(z), \end{aligned}$$
(21)

where the polynomial function

$$\begin{aligned} \varphi _\ell (z)= P_\ell \left( \frac{2(z-z_m)}{\Delta z_m}\right) ,\, \ell =1,2,3,\, \ldots , p. \end{aligned}$$
(22)

If \(p=0\), the piecewise constant basis functions are used in the approximate solution \(w_k\). If \(p=1\), \(w_k\) uses linear basis functions. The linear basis functions are utilized in the current study, and thus \(\ell\) = 0, 1. Furthermore,

$$\begin{aligned}&w_m^{\ell }(t)= \frac{2\ell +1}{\Delta z_{m}} \int _{\Omega _m} w_k(t,z) \varphi _\ell (z)dz,\quad \nonumber \\&g_m^{\ell }(t)= \frac{2\ell +1}{\Delta z_{m}} \int _{\Omega _m}g_k(c_k) \varphi _\ell (z)dz. \end{aligned}$$
(23)

In this formulation, the exact solutions w and g are substituted by their approximate counterparts \(w_k\) and \(g_k\) respectively. Similarly the function u(z) is replaced by the test function \(\varphi _\ell \in U_K\). Here \(\{\varphi _\ell \}_{\ell =1}^{N_{p}-1}\) serves as a suitable basis for \(U_k\). Furthermore, at the cell interface \(z_{m+\frac{1}{2}}\), the function \(f(c_{m+\frac{1}{2}},g_{m+\frac{1}{2}})=f(c(t,z_{m+\frac{1}{2}}),g(c_{m+\frac{1}{2}}))\) is not defined. As a result, it must be supplanted by a numerical flux centered on two \(c_k(t,z)\) points at the discontinuity i.e.,

$$f\Big(c_{m+\frac{1}{2}},g_{m+\frac{1}{2}}\Big)\approx k_{m+\frac{1}{2}}=k\Big(c_{m+\frac{1}{2}}^{-},c_{m-\frac{1}{2}}^{+}\Big).$$
(24)

Since \(g:= g(c)\), it can be removed from the arguments of k for simplification. Then,

$$\begin{aligned}&c_{m+\frac{1}{2}}^{-}:=c_k {\Big(t,z_{m+\frac{1}{2}}^{-}\Big)}=\sum _{\ell =0}^{p}{c_{m}^{\ell }} \varphi _{\ell }\Big(z_{m+\frac{1}{2}}\Big), \quad \nonumber \\&c_{m-\frac{1}{2}}^{+}:=c_k {\Big(t,z_{m-\frac{1}{2}}^{+}\Big)}=\sum _{\ell =0}^{p}{c_{m}^{\ell }} \varphi _{\ell }\Big(z_{m-\frac{1}{2}}\Big). \end{aligned}$$
(25)

The weak formulations in Eqs. (18) and (19) are simplified to:

$$\frac{dw_j^{\ell }(t)}{dt}=-\frac{2\ell +1}{\Delta z_{m}}\left( \varphi _\ell \Big(z_{m+\frac{1}{2}}\Big)k_{m+\frac{1}{2}}- \varphi _\ell \Big(z_{m-\frac{1}{2}}\Big) k_{m-\frac{1}{2}}\right)+\frac{2\ell +1}{\Delta z_{m}} \int _{\Omega _m}f(c_k,g_k)\frac{d\varphi _\ell (z)}{dz}dz,$$
(26)
$$g_m^{\ell }(t)=\frac{2\ell +1}{\Delta z_{m}}\sqrt{D_{z,i}(\phi )}\left( \varphi _\ell \Big(z_{m+\frac{1}{2}}\Big) c_{m+\frac{1}{2}}-\varphi _\ell \Big(z_{m-\frac{1}{2}}\Big) c_{m-\frac{1}{2}}\right) -\frac{2\ell +1}{\Delta z_{m}}\sqrt{D_{z,i}(\phi )}\int _{\Omega _m}c_k(t,z)\frac{d\varphi _\ell (z)}{dz}dz.$$
(27)

The liquid concentration c is required at each time step to update the isotherm function \(q^*(c)\) and flux function f(c, g(c)). Meanwhile, in each mesh interval \(\Omega _m\), Eq. (26) gives the modified value of \(w_m=c_m(t)+Fq^{*}_{m}(c)\). To handle this problem, we can use the chain rule and the system of ODE can be achieved in terms of \(c_m\). Thus, the scheme in Eq. (26) turns out to be

$$\begin{aligned}&\frac{d{c_m^{\ell }(t)}}{dt}=-\left[ 1+\frac{d\left( F(q^{*})_m^{(\ell )}(c)\right) }{d{c_m^{\ell }(t)}} \right] ^{-1}\left( \frac{2\ell +1}{\Delta z_{m}}\right) \nonumber \\&\left( \varphi _\ell \Big(z_{m+\frac{1}{2}}\Big)k_{m+\frac{1}{2}}- \varphi _\ell \Big(z_{m-\frac{1}{2}}\Big) k_{m-\frac{1}{2}}\right) \nonumber \\&+\left[ 1+\frac{d\left( F(q^{*})_m^{(\ell )}(c)\right) }{d{c_m^{\ell }(t)}}\right] ^{-1}\left( \frac{2\ell +1}{\Delta z_{m}}\right) \int _{\Omega _m}f(c_k,g_k)\frac{d\varphi _\ell (z)}{dz}dz. \end{aligned}$$
(28)

Similarly, the scheme in Eq. (27) is of the following form

$$g_m^{\ell }(t)=\frac{2\ell +1}{\Delta z_{m}}\sqrt{D_{z,i}(\phi )} \left( \left( \varphi _\ell \Big(z_{m+\frac{1}{2}}\Big) c_{m+\frac{1}{2}}-\varphi _\ell \Big(z_{m-\frac{1}{2}}\Big) c_{m-\frac{1}{2}}\right) \right) -\frac{2\ell +1}{\Delta z_{m}}\sqrt{D_{z,i}(\phi )} \left( \int _{\Omega _m}c_k(t,z)\frac{d\varphi _\ell (z)}{dz}dz\right) .$$
(29)

The initial data from Eq. (23) for the system described above are as follows:

$$\begin{aligned} c_m^{\ell }(0)= \frac{2\ell +1}{\Delta z_{m}} \int _{\Omega _m} c(0,z) \varphi _\ell (z)dz,\quad g_m^{\ell }(0)=g(c_m^{\ell }(0)). \end{aligned}$$
(30)

The choice of appropriate numerical flux is still remaining. The local Lax–Friedrichs flux is utilized in this study because of its simplicity and accuracy:

$$\begin{aligned} k^{LLF}(a,b)=\frac{1}{2}[h(a,g(a))+h(b,g(b))+C(b-a)], \end{aligned}$$
(31)
$$\begin{aligned} \max _{min(a,b)\le s \le max(a,b)}|h^{'}{(s,g(s))}|. \end{aligned}$$
(32)

The Gauss–Lobatto quadrature method of order 10 is utilized for approximating the integral terms arising in Eqs. (29) & (30).

To avoid numerical oscillations, some restricting methods must be implemented for attaining local monotonicity. Therefore, it is much necessary to modify \(c^{\pm }_{m+\frac{1}{2}}\) in the Eq. (24). Then, Eq. (25) can be expressed as

$$\begin{aligned} c^{-}_{m+\frac{1}{2}}={c}^{0}_{m}+\hat{c}_{m}, \quad c^{+}_{m-\frac{1}{2}}={c}^{0}_{m}-\check{c}_{m}, \end{aligned}$$
(33)

where

$$\begin{aligned} \hat{c}_{m}=\sum _{\ell =1}^{m}{c_{m}^{\ell }} \varphi _{\ell }\Big(z_{m+\frac{1}{2}}\Big), \quad \check{c}_{m}=-\sum _{\ell =1}^{m}{c_{m}^{\ell }} \varphi _{\ell }\Big(z_{m-\frac{1}{2}}\Big). \end{aligned}$$
(34)

Here, we have taken the linear basis functions such that \(\ell =1,0\). Thus, \(\hat{c}_{m}\) and \(\check{c}_{m}\) can be transformed as

$$\begin{aligned}&\hat{c}_{m}^{(mod)}=mm\big(\hat{c}_{m},\Delta +{c}_{m}^{(0)},\Delta -{c}_{m}^{(0)}\big),\nonumber \\&\check{c}_{m}^{(mod)}=mm\big(\check{c}_{m},\Delta +{c}_{m}^{(0)},\Delta -{c}_{m}^{(0)}\big), \end{aligned}$$
(35)

where \(\Delta ^{\pm }:=(c_{m\pm \frac{1}{2}}-c_{m})\) and the minmod function is denoted mm. Equation (33) can be replaced by

$$\begin{aligned} c^{-{(mod)}}_{m+\frac{1}{2}}={c}^{(0)}_{m}+\hat{c}_{m}^{(mod)}, \quad c^{+{(mod)}}_{m-\frac{1}{2}}={c}^{(0)}_{m}-\check{c}_{m}^{(mod)}. \end{aligned}$$
(36)

Then, Eq. (24) can be modified to

$$k_{m+\frac{1}{2}}=k\left(c_{m+\frac{1}{2}}^{-(mod)},c_{m-\frac{1}{2}}^{+(mod)}\right).$$
(37)

The accuracy is not affected in those smooth sections and the convergence can be obtained without oscillations in the vicinity of the shock. Ultimately, a Runge–Kutta technique preserving the TVB property is required to solve the obtained ODE-system. On the right end of the column, an outflow boundary condition is used \(c^{\ell }_{N+1}= c^{\ell }_{N}\).

4 Numerical case studies

This section outlines some numerical experiments to scrutinize the behaviour of gradient profiles on various key parameters such as \(k_{Hr}\), \(\alpha\), \(\gamma\), \(D_{zr}\) and \(b^\textrm{ref}\) characterizing the reference Henry’s constant, the component specific solvent strength parameter, the reference axial dispersion coefficient, and the reference nonlinearity extent coefficient, respectively. In the interest of clarity, the axial dispersion coefficients \(D_\textrm{z,i}=D_\textrm{zr,i}\), and the Henry’s constant \(K_\textrm{H,i}=k_\textrm{Hr,i}\). At \(z=L\) the solute concentration (c) and the modulator concentration \(\phi\) are plotted versus time t. The concentration profile generated by the DG-FE method is compared to the concentration profile generated by the HR-FVM method. The DG scheme is observed to be more accurate than other flux-limiting FV techniques. In these case studies, the DG scheme enhanced the resolution of the problems. All the required parameters used in the numerical test studies are selected in compliance with commonly used ranges in HPLC applications. The values of optimal parameters occurring in the model equations are catalogued in Tables 1 and  2.

Table 1 Reference parameters used in simulations for single-component
Full size table
Table 2 Reference parameters used in simulations for two-component
Full size table

4.1 Elution of single component

Few examples of non linear single component elution are analyzed and presented below:

Figure 1a illustrates the influence of varying Henry’s constant, from \(k_{Hr}=5\) to \(k_{Hr}=9\), using a linear isotherm model, while keeping the solvent strength parameter, \(\alpha\), fixed at 0.9. The figure highlights changes in peak shapes as \(k_{Hr}\) increases, demonstrating distinct variations in their profiles. In Fig. 1b, the effect of Henry’s constant on the non-linear concentration profiles is explored with a reference value of \(b^\textrm{ref}=2\). This scenario produces sharp, asymmetrical fronts in the peaks, along with significant differences between them. Additionally, a comparison is made between the Discontinuous Galerkin Finite Element (DG-FE) and Finite Volume (FV) schemes for each value of \(k_{Hr}\). The results clearly show that the DG method yields more accurate solutions, especially at points of sharp discontinuities, compared to the high-resolution finite volume scheme. Figure 1c examines how variations in the solvent strength parameter, \(\alpha\), affect the nonlinear single-component elution profile when \(k_{Hr}\) is held constant at 5.0. The findings indicate an increasing sensitivity of the peak profiles to changes in the modulator concentration, influenced by the reference Henry constant.

Fig. 1
figure 1

Comparison of DG-scheme and FVS for the effects of Henry’s constant \(k_{Hr}\) on a linear and b nonlinear single-component elution profile. c Effects of \(\alpha\) on non-linear single-component elution profile. Other parameters are catalogued in Table 1

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Figure 2 demonstrates how varying Henry’s constant, \(k_{Hr}\), impacts the temporal moments derived from concentration profiles. These moments effectively predict the behavior observed in the concentration profiles. Specifically, the graphs of the first moments show a clear linear relationship, indicating that the mean retention times increase consistently as the Henry constant values rise. Furthermore, when examining higher values of \(k_{Hr}\), the representations of the second and third central moments reveal noticeable deviations. These deviations highlight more complex dynamics in peak dispersion and asymmetry, suggesting that higher Henry constants influence not only the retention times but also the spread and shape of the peaks in more pronounced ways.

Fig. 2
figure 2

Effect of reference Henry’s constant \(k_{Hr}\) on temporal moments for \(t_s=5\) min, \(t_e=80\) min, \(\phi _0=0.1\) and \(\phi _e=0.95\)

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Figure 3a illustrates the impact of varying the axial dispersion coefficient, \(D_{zr}\), from 0.1 to 0.0002 cm\(^{2}\)/min in a linear isotherm scenario, holding \(\gamma\) at 1.0 and the reference nonlinearity coefficient, \(b^\textrm{ref}\), at 0. As \(D_{zr}\) decreases, the chromatographic peaks become more sharply defined and narrower, although the retention times at the peak maximum heights remain constant.

In Fig. 3b, a similar trend is observed under nonlinear isotherm conditions. A comparative analysis between the Discontinuous Galerkin scheme (DG-scheme) and Finite Volume Scheme (FVS) indicates that lower values of \(D_{zr}\) result in steeper, asymmetrical peak fronts. These characteristics are more accurately resolved by the DG-scheme. Notably, in the nonlinear isotherm case, the reduction in \(D_{zr}\) correlates with decreased retention times.

Figure 3c explores the influence of \(\gamma\), which represents the sensitivity of the axial dispersion coefficient to solvent strength, across a range from \(-\,4.0\) to 2.0. The findings demonstrate that the responsiveness of the axial dispersion coefficient to variations in solvent strength progressively increases, affecting the nonlinear elution profiles significantly. These results underline the critical role of solvent strength in modulating dispersion within chromatographic systems.

Fig. 3
figure 3

Comparison of DG-scheme and FVS for the effects of reference axial dispersion coefficient \(D_{zr}\) on a linear and b nonlinear single-component elution profile. c Effects of \(\gamma\) on nonlinear single-component elution profile. Other parameters are catalogued in Table 1

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Figure 4 presents the influence of the axial dispersion coefficient, \(D_{zr}\), on various temporal moments. Analysis of the first moment reveals that an increase in \(D_{zr}\) corresponds to a proportional increase in retention time. This relationship underscores the direct impact of dispersion on solute migration through the chromatographic column.

Further examination through the second central moment indicates that the dispersion of the solute profiles broadens as \(D_{zr}\) increases. This broadening effect highlights the enhanced spread of solute distribution within the column due to increased dispersion.

Additionally, the third central moment’s plots demonstrate an increase in skewness with rising values of \(D_{zr}\). This increase suggests that higher dispersion coefficients lead to more pronounced asymmetry in peak shapes.

It is noteworthy that the values derived from these moment calculations align well with those obtained through numerical integration of the elution profiles. This congruence validates the mathematical expressions used for moment calculations and confirms the accuracy of the numerical integration technique in capturing the dynamics of axial dispersion within the chromatographic system.

Fig. 4
figure 4

Effect of reference axial dispersion coefficient \(D_{zr}\) on temporal moments. Here \(t_s=5\) min, \(t_e=80\) min, \(\phi _0=0.1\) and \(\phi _e=0.95\)

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Figure 5a illustrates the impact of the reference nonlinearity coefficient, \(b^\textrm{ref}\), on the elution profiles, maintaining a constant solvent strength parameter, \(\alpha =0.9\). At a coefficient value of \(b^\textrm{ref}=0\), the elution profile assumes a Gaussian shape. As the value of \(b^\textrm{ref}\) increases, the retention time correspondingly decreases, and the elution peaks exhibit increased tailing. The saturation capacity of the stationary phase decreases. This reduction in saturation capacity significantly alters the isotherm shape, which in turn impacts the peak shape observed in chromatographic elution profiles. Additionally, a comparative analysis between the Discontinuous Galerkin (DG) scheme and the high-resolution Finite Volume (HR-FV) scheme is presented as the nonlinearity coefficient varies from 0 to 2. The results indicate that the DG-scheme consistently outperforms the HR-FV scheme, especially at points of acute discontinuities, providing more accurate resolutions.

In Fig. 5b, the influence of \(\gamma\) on the nonlinear elution profiles is explored, with \(b^\textrm{ref}\) fixed at 2.0. As \(\gamma\) increases from \(-\) 2.0 to 2.0, both the peak heights and the retention times of the profiles are observed to increase. This trend highlights the sensitivity of the system’s response to changes in \(\gamma\), affecting the overall dynamics and efficiency of the separation process.

Fig. 5
figure 5

a Comparison of DG-scheme and FVS of nonlinearity coefficient \(b^\textrm{ref}\) on linear single-component elution. b Influence of \(\gamma\) on nonlinear single-component elution profile. Other parameters are catalogued in Table 1

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The influence of the nonlinearity coefficient \(b^\textrm{ref}\) on moments is depicted in Fig. 6. The plot of the first moment revealed that the retention time decreases with an increase in the values of \(b^\textrm{ref}\). The plot of the second moment indicates that the variance of the profiles increases with increments in the \(b^\textrm{ref}\) values. The plot of the third moment depicts that skewness of the fronts significantly improved with higher values of \(b^\textrm{ref}\). Once again, the values computed directly by moment expressions correlate well with those acquired through numerical integration of the elution profiles.

Fig. 6
figure 6

Effect of nonlinearity coefficient \(b^\textrm{ref}\) on temporal moments for \(t_s=5\) min, \(t_e=80\) min, \(\phi _0=0.1\) and \(\phi _e=0.95\)

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Figure 7 (left plot) illustrates the effects of varying the gradient start time (\(t_s\)) while maintaining a constant gradient end time (\(t_e = 80\) min) on the nonlinear elution profiles. In the case where \(t_s = t_e = 80\) min, a distinct scenario unfolds in which a rapid transition to the isocratic state occurs shortly thereafter, with both initial and final solvent strengths (\(\phi _0\) and \(\phi _e\)) set at 0.9. Introducing steeper gradients by delaying \(t_s\) leads to broader peaks with reduced heights and extended tails, as well as a deceleration in the chromatographic process. This effect continues until a critical steepness is reached, beyond which the elution is predominantly governed by the initial solvent strength. An inflection point in gradient starting time is observed between 20 and 40 min.

In Fig. 7 (right plot), the impact of the gradient end time (\(t_e\)) on nonlinear elution profiles is demonstrated. When \(t_s = t_e = 5\) min, there is a swift shift to an isocratic state characterized by \(\phi _0 = 0.1\) and \(\phi _e = 0.9\). The critical transition point for \(t_e\) is identified to occur between 20 and 30 min. This analysis highlights the significant role of both gradient initiation and termination times in shaping the dynamics of the chromatographic separation process.

Fig. 7
figure 7

Left plot: Effect of \(t_s\) on elution profiles for \(t_e=80\) min, \(\phi _0=0.1\) and \(\phi _e=0.95\). Right plot: Effect of \(t_e\) on elution profiles for \(t_s=5\) min, \(\phi _0=0.1\) and \(\phi _e=0.95\)

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4.2 Elution of two-component

In Fig. 8a–c, we explore the influence of the solvent strength parameter, \(\alpha\), on nonlinear elution profiles, where \(\alpha\) ranges from 0.9 to 3.0. The reference Henry constants \(k_{Hr,1} = 4.5\) and \(k_{Hr,2} = 9.0\) remain constant throughout the analysis. As \(\alpha\) increases, there is a noticeable decrease in retention times accompanied by an increase in peak heights for the two components. This pattern indicates that higher solvent strength facilitates faster elution and enhances solute peak prominence.

Conversely, Fig. 9a–c examine the effects of varying \(\gamma\), the parameter representing the sensitivity of the axial dispersion coefficient to changes in solvent strength, which increases from \(-\) 2.0 to 2.0. The axial dispersion coefficient is held constant at \(D_{zr} = 0.0002\,\text{cm}^2/\text{min}\). It is observed that variations in \(\gamma\) do not significantly impact the average retention times of the profiles, suggesting that changes in this parameter primarily affect other aspects of the elution profile, such as peak dispersion and shape, rather than the overall elution timing. This distinction underscores the different roles that solvent strength and dispersion sensitivity play in the dynamics of chromatographic separation.

Fig. 8
figure 8

Comparison of DG-scheme and FVS for the effects of gradually changing \(\alpha\) on non-linear two-component elution profiles. Other parameters are catalogued in Table 2

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Fig. 9
figure 9

Comparison of DG-scheme and FVS for the effects of gradually changing \(\gamma\) on non-linear two-component elution profiles. Other parameters are catalogued in Table 2

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Figure 10 presents the outlet profiles of two components under both isocratic and gradient elution conditions, highlighting scenarios where the concentration of modulators varies significantly. The experimental parameters utilized include an axial dispersion coefficient \(D_{zr} = 0.0002\, \text{cm}^2/\text{min}\), reference nonlinearity coefficients \(b_{1}^\textrm{ref} = 1.0\) and \(b_{2}^\textrm{ref} = 2.0\), Henry constants \(k_{Hr,1} = 4.5\) and \(k_{Hr,2} = 9.0\), with additional parameters sourced from Table 2.

In isocratic elution, characterized by a constant solvent strength \(\phi _0 = \phi _e\), the resulting profiles for the nonlinear two-component system are broader, exhibiting significant variances and asymmetries, as well as extended retention times. This outcome suggests a pronounced interaction of the components with the stationary phase under a constant solvent environment, leading to diverse elution dynamics.

Conversely, the profiles obtained from gradient elution are notably more condensed, with reduced retention times. This difference indicates that the varying solvent strength in gradient elution facilitates a more efficient separation process, reducing the interaction time with the stationary phase and leading to sharper, more symmetrical peaks. This comparative analysis underscores the impact of elution mode on the behavior and resolution of chromatographic peaks in multi-component systems.

Fig. 10
figure 10

Comparison of isocratic and gradient elution considering non-linear isotherms on non-linear two-component elution profile. Other parameters are catalogued in Table 2

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5 Conclusion

The variability of solvent retention activity in gradient elution, influenced by changes in the composition of the mobile phase, renders the mathematical solutions for gradient elution models significantly more complex than those for isocratic elution. This research explored the development and characteristics of elution profiles across various parameters, including modulator concentration, gradient initiation and termination times, gradient slope, and solvent strength. To solve the model equations for both single and dual-component elution, a discontinuous Galerkin (DG-FEM) scheme was employed. This scheme adheres to the Total Variation Bounding (TVB) property and achieves second-order accuracy. Moreover, the scheme offers flexibility for enhancement to higher order accuracies through the incorporation of higher order basis functions and advanced slope limiters, such as Weighted Essentially Non-Oscillatory (WENO) limiters. The efficacy of the DG-scheme was corroborated through comparative analyses with other Finite Volume (FV) schemes documented in the literature. Notably, the DG-scheme demonstrated superior performance over the high-resolution Koren scheme in numerical test scenarios, particularly in resolving sharp discontinuities and narrow peaks effectively. Gradient elution chromatography is found to be beneficial when there is a significant variation in the retention times of components. In the future, we plan to integrate temperature gradient and solvent gradient techniques to gain further control over the flow rates of analytes within the chromatographic column. This integration will allow us to more precisely manipulate the speed of pulses, enhancing separation efficiency and process optimization.