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Compact implicit high-resolution numerical scheme for multicomponent transport problems with Langmuir sorption

Dagmar Zakova, Peter Frolkovic

arXiv:2609.08320v1math.NAcs.NAmath.AP

TL;DR

Nonlinear Langmuir sorption couples multicomponent transport and complicates explicit discretization. The paper develops a compact implicit high-resolution scheme for these systems. Its experiments report accuracy for smooth solutions, stability at larger Courant numbers than explicit schemes, and capture of sharp chromatography fronts.

  • Problem

    Nonlinear multicomponent Langmuir sorption creates a coupled transport system that standard component-wise explicit discretizations do not respect.

    Source: §1 · p. 2, §6 · p. 21

  • Method

    A compact implicit high-resolution scheme uses local algebraic solves for Langmuir-coupled transport and considers characteristic Courant numbers in the limiter.

    Source: §1 · p. 2, §6 · p. 21

  • Results

    The schemes achieve expected accuracy for smooth solutions and remain stable at Courant numbers significantly larger than those admissible for explicit schemes; chromatography examples capture sharp moving fronts and displacement trains.

    Source: §6 · pp. 21–22

  • Takeaways & Limitations

    In multicomponent Langmuir systems, the method captures sharp moving fronts and displacement trains in chromatography examples.

    Source: §6 · p. 22

    The study treats only the advective part of transport and omits diffusion-dispersion processes.

    Source: §1 · p. 1

Abstract

from arXiv · show

We present a compact implicit high-resolution finite volume scheme for one-dimensional multicomponent transport problems with nonlinear Langmuir sorption. The method combines a second order compact implicit discretization with WENO reconstruction and a time limiter for the treatment of steep gradients and discontinuities. A special attention is given to the Courant number used in the limiter, where local characteristic Courant numbers based on the eigenvalues of the inverse retardation matrix are considered. The compact implicit schemes are compared with analogous explicit first and second order methods. Numerical experiments show second order convergence for smooth solutions and demonstrate that the implicit approach can be used with significantly larger Courant numbers. The method is also applied to a three-component displacement chromatography problem with sharp concentration fronts.

1 Introduction

Nonlinear sorption creates tightly coupled transport equations that standard component-wise explicit discretizations do not respect. The paper develops a compact implicit high-resolution scheme using local solves to handle this coupling and steep profiles while allowing larger time steps.

1 Introduction

  • The motivating applications include groundwater contaminant transport and preparative liquid chromatography, while this study treats advection and omits diffusion-dispersion.The omitted processes are referred to standard approaches for their inclusion.Source: §1 · p. 1
  • Nonlinear accumulation from sorption forces local nonlinear solves in explicit methods, while component-wise discretizations fail to respect coupling across the system.The paper frames this tight coupling as a central challenge for multicomponent transport.Source: §1 · p. 2
  • The compact implicit method uses small local algebraic systems in each grid cell while respecting the transport-reaction equations’ tight coupling.This avoids solving large, globally coupled nonlinear systems.Source: §1 · p. 2
  • For smooth solutions, the scheme is second order accurate and can use larger time steps when fast solutes leave the domain but slower components remain.Unlike explicit schemes, its time step need not follow the fastest component after that component exits the domain.Source: §1 · p. 2
  • The high-resolution extension combines local stencil adjustment with limiting toward first order accuracy to control oscillations near discontinuities and when time steps violate the CFL condition.The stencil adjustment is analogous to WENO approximations.Source: §1 · p. 2

2 Mathematical model

The model first describes scalar porous-medium transport with sorption, then extends it to coupled concentrations whose characteristic speeds depend on a nonlinear retardation matrix.

2.1 Scalar transport equation with sorption

  • The scalar model tracks mobile concentration in a porous medium, while total concentration includes both mobile and adsorbed phases.The model considers Langmuir sorption; its parameters characterize adsorbed amount and how rapidly adsorption approaches saturation.Source: §2.1 · p. 3
  • Sorption retards transport: stronger retardation corresponds to smaller characteristic speeds, and the model links retardation to adsorption and porosity.With weak sorption, the characteristic speed approaches the transport velocity.Source: §2.1 · p. 4

2.2 Extension to a system of equations

  • The system extension transports multiple mobile concentrations at the same velocity and writes each component’s balance using its total concentration.The total concentration is defined component-wise for the multicomponent Langmuir model.Source: §2.2 · p. 4
  • The Jacobian of the total-concentration mapping acts as a nonlinear retardation matrix, and its inverse determines the system’s characteristic speeds.Larger eigenvalues of the Jacobian correspond to stronger retardation and slower wave propagation; eigenvalues near one correspond to weaker retardation and faster transport.Source: §2.2 · p. 5

3 Second order accurate schemes

The section develops compact implicit finite-volume schemes, including a second-order formulation and a high-resolution extension, then sets them against first- and second-order explicit methods. The explicit schemes retain local nonlinear recovery, and the second-order version also has a wider stencil and a stability-limited time step.

3.1 Compact implicit scheme

  • The second-order compact implicit scheme replaces the interface value with a compact approximation using new- and previous-time values, while its resulting system contains only new-time unknowns.The first-order implicit upwind scheme instead uses a simple interface approximation and is robust but significantly diffusive near sharp fronts.Source: §3.1 · p. 6
  • For nonsmooth solutions, the high-resolution implicit form varies reconstruction parameters by component and cell, with WENO reconstruction and a time limiter controlling the second-order correction.WENO favors locally smooth stencils; the limiter can reduce the correction toward first-order upwind near steep fronts or discontinuities, particularly at large Courant numbers.Source: §3.1 · p. 6

3.2 Explicit schemes

  • The second-order explicit scheme uses a Lax–Wendroff-type time discretization adapted to nonlinear accumulation, with Jacobian terms and reconstructed interface values.The explicit update still requires local Newton solves to recover concentrations from accumulated concentrations because of the nonlinear relation F(C).Source: §3.2 · pp. 7–8
  • The second-order explicit method has a wider stencil, still requires local Newton iterations, and remains stability-limited, making it unsuitable for the large Courant numbers used in the experiments.Additional neighboring or extrapolated ghost values are needed near boundaries.Source: §3.2 · p. 8

4 High-resolution schemes

The high-resolution compact implicit scheme combines WENO reconstruction with a time limiter, whose Courant number can reflect local characteristic speeds. The explicit WENO variant uses componentwise reconstruction parameters and omits time limiting under its stability-restricted time steps.

4.1 High-resolution compact implicit scheme

  • The implicit scheme computes WENO reconstruction parameters and time-limiter values separately for each concentration component.The time limiter matters especially for large time steps and enters the condition used to preserve non-oscillatory behavior.Source: §4.1 · p. 8
  • A global upper-bound Courant number can over-limit some components, increasing numerical diffusion and potentially reducing accuracy when sorption makes characteristic speeds much smaller than velocity u.The characteristic values are based on eigenvalues of J(C)^-1.Source: §4.1 · p. 9
  • Using local characteristic Courant numbers from eigenvalues of J(C)^-1 makes the limiter less restrictive than the global upper bound when sorption slows the characteristic speeds.The time step can still be chosen using the global bound, while the limiter uses local characteristic values to avoid unnecessarily reducing the second-order correction.Source: §4.1 · p. 9

4.2 WENO approximation in explicit scheme

  • In the explicit scheme, the componentwise reconstruction parameter may be fixed or adaptively WENO-selected from previous time levels, and no time limiter is applied under its stability-restricted time steps.In smooth regions with comparable differences, the parameter remains close to 1/2.Source: §4.2 · p. 10

5 Numerical experiments

The numerical experiments assess convergence, efficiency, stability at large Courant numbers, and sharp-front capture across smooth Langmuir, linear benchmark, and displacement chromatography problems. They include chromatography computations examining front resolution and how a lower displacer concentration affects displacement, separation, and formation of the isotachic train.

5 Numerical experiments

  • The experiments measure final-time discrete L1 error component-wise and sum across system components; EOC is computed from consecutive mesh refinements, and Langmuir cases report a global Courant upper bound alongside local characteristic values.The local characteristic Courant numbers do not exceed the global bound in the Langmuir cases considered and are often smaller.Source: §5 · p. 10

5.1 First experiment: smooth three-component Gaussian profiles

  • Mesh refinement confirms expected convergence; at Comax = 0.5, implicit constant-ω schemes have smaller errors than explicit counterparts, while WENO errors are comparable and slightly favor explicit schemes on finer meshes.At Comax = 8, implicit schemes remain stable and preserve expected EOCs on sufficiently fine grids, with the WENO time limiter active to control corrections and suppress spurious oscillations.Source: §5.1 · pp. 12–13, Table 1 · p. 12, Table 2 · p. 13

5.2 Second experiment: Linear benchmark problem

  • In these two experiments, implicit schemes used CoL values of 12.5, 25, 50, 100, 200, 400, and 800; these large values are admissible for implicit schemes but not explicit schemes under their stronger stability restriction.The two characteristic Courant numbers satisfy CoL = 100 CoS.Source: §5.2 · p. 14
  • CoL = 100 gives the best overall balance of error and CPU time for the smooth linear benchmark with the implicit constant-ω second-order scheme, while explicit schemes become unstable at the larger tested Courant numbers.The selected CoL gives the smallest errors among tested Courant numbers while requiring reasonable CPU time.Source: §5.2 · pp. 14, 16, Table 3 · p. 15, Figure 3 · p. 15
  • With CoL = 800, space limiting alone leaves visible oscillations; time limiting based on CoL controls them but increases errors, whereas limiting on CoS yields smaller errors and more accurate profiles while controlling oscillations.The fast characteristic component is absent in this discontinuous test, so using CoS avoids unnecessary limiting; Figure 6 shows sharper, less diffusive profiles than with CoL.Source: §5.2 · pp. 16–17, Table 4 · p. 17, Figure 6 · p. 18

5.3 Third experiment: three-component displacement chromatography

  • 4.80 was the largest final-time local characteristic Courant number, below the global bound Comax = 12 but above the stability limit usually required by explicit schemes.The global bound was not used directly because it would cause excessive time limiting and push the method too strongly toward first-order accuracy.Source: §5.3 · p. 19
  • The final chromatography profiles separate the three components into distinct zones with increasingly sharp interfaces as the displacer pushes the sample components through the column.A second simulation keeps the domain, mesh, final time, and method unchanged while reducing the continuously injected displacer concentration from 1 to 0.5.Source: §5.3 · p. 20, Figure 7 · p. 20
  • At Comax = 12, the compact implicit high-resolution scheme produced well-resolved concentration fronts without visible oscillations.Source: Figure 8 · p. 21, §5.3 · p. 21
  • Figure 8 shows profiles for the second inlet at t ≈ 1, 4, 8, and 15, with u = 20, c3 = 0.5, and Comax = 12.Source: Figure 8 · p. 21
  • With c3 = 0.5, displacement was weaker and separation less pronounced; the isotachic train did not fully form within the interval, consistent with [4].Source: §5.3 · p. 21

6 Conclusions

The multicomponent Langmuir isotherm creates nonlinear coupling, a nonlinear retardation matrix, and multiple characteristic speeds. The compact implicit schemes achieve expected accuracy for smooth solutions, remain stable at larger Courant numbers, and capture sharp chromatography fronts and displacement trains.

6 Conclusions

  • Multicomponent Langmuir sorption creates nonlinear coupling, a nonlinear retardation matrix, and a hyperbolic system with several characteristic speeds.Source: §6 · p. 21
  • The schemes achieve expected accuracy for smooth solutions and remain stable at Courant numbers significantly larger than those admissible for explicit schemes.Explicit schemes still require nonlinear inversion of the accumulation function and face a stronger stability condition.Source: §6 · p. 21
  • Characteristic Courant numbers based on eigenvalues of J(C)^−1 reduce unnecessary numerical diffusion and help preserve more of the scheme's second-order accuracy.The global bound Comax = τu/h may cause excessive limiting when sorption reduces actual characteristic speeds.Source: §6 · p. 21
  • The chromatography examples capture sharp moving fronts and displacement trains in multicomponent Langmuir systems.Source: §6 · p. 22
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