[Paper Review] Numerical modelling of convection-diffusion-reaction problems with free boundary in 1D

Gabriela Kacurova|ArXiv.org|Sep 2, 2009

Fluid Dynamics and Thin Films7 references3 citations

TL;DR

This paper presents a novel numerical method for 1D convection-diffusion-reaction problems with free boundaries, modeling the interface evolution via an ODE derived from local traveling wave profiles. By transforming the problem to a fixed domain and using adaptive spatial discretization near the interface, the method achieves high accuracy in capturing sharp fronts, validated through comparisons with analytical solutions and prior numerical studies for porous media and reaction-diffusion equations.

ABSTRACT

We discuss a numerical method for convection-diffusion-reaction problems with a free boundary in 1D. The method is based on the numerical modelling of the interface evolution, the transformation to a fixed domain problem and the approximation by an ODE system. The interface evolution is modelled by means of the local shape of the corresponding travelling wave solution. The method can be applied to many free boundary problems with a finite speed of the interface. The presented method can also approximate some problems with an infinite speed of the interface for damped travelling wave type solutions. In the numerical experiments we compare our numerical solution with the analytical ones for some problems.

Motivation & Objective

• To develop an efficient and accurate numerical method for 1D convection-diffusion-reaction problems with free boundaries, particularly those with finite-speed interfaces.
• To model the time evolution of the interface $ s(t) $ using a local ODE derived from traveling wave solutions, avoiding expensive adaptive meshing.
• To transform the moving-boundary problem into a fixed-domain problem via $ y = x/s(t) $, enabling stable and accurate spatial discretization.
• To apply the method to degenerate diffusion, porous media, and reaction-diffusion equations, including cases with non-degenerate and damped traveling wave solutions.
• To validate the method against analytical solutions and existing numerical results for benchmark problems such as the porous media equation and turbulent flow models.

Proposed method

• Model the interface evolution $ s(t) $ via an ODE derived from the local shape of traveling wave solutions, specifically $ \dot{s}(t) = -\frac{n}{n-1} \partial_x u^{n-1} \big|_{x \nearrow s(t)} $ for porous media-type equations.
• Transform the moving domain $ (0, s(t)) $ to a fixed reference domain $ (0, 1) $ using the change of variables $ y = x/s(t) $, enabling standard spatial discretization.
• Apply a specialized spatial discretization with higher resolution near $ y = 1 $ (i.e., $ x = s(t) $) to capture the sharp front accurately.
• Discretize the transformed PDE into a system of ODEs in time, including the interface ODE, forming a stiff ODE system solvable by robust numerical solvers.
• Use the method to approximate non-degenerate cases (e.g., linear sorption) by perturbing the nonlinearity with $ p \nearrow 1 $, e.g., $ \Psi(s^p) $, to avoid interface formation while maintaining accuracy.
• Validate results by comparing numerical solutions and interfaces with analytical solutions (e.g., Barenblatt-Pattle) and published numerical benchmarks.

Experimental results

Research questions

• RQ1 Can the interface evolution in 1D convection-diffusion-reaction problems be accurately modeled using a local ODE derived from traveling wave profiles?
• RQ2 Does transforming the moving-boundary problem to a fixed domain via $ y = x/s(t) $ enable stable and high-accuracy numerical solution with adaptive spatial resolution near the interface?
• RQ3 Can the method accurately approximate both degenerate (finite-speed interface) and non-degenerate (infinite-speed, damped wave) cases, including sorption isotherms?
• RQ4 How does the method compare in accuracy and efficiency to existing methods for benchmark problems such as the porous media equation and turbulent flow models?
• RQ5 Can the method be extended to approximate non-degenerate problems by perturbing the nonlinearity with $ p \nearrow 1 $, and how stable is this approximation?

Key findings

• The method successfully captures the sharp front of the solution with high accuracy, as evidenced by the near-perfect overlap between numerical and analytical interfaces in the porous media case ($ n=6 $).
• For the porous media equation, the numerical interface $ s(t) $ matches the analytical solution $ s(t) = [\frac{2n(n+1)}{n-1}(t+1)]^{1/(n+1)} $ with negligible error, confirming the validity of the interface ODE model.
• In the case of non-degenerate diffusion with $ p=0.95 $, the method provides a stable and accurate approximation of linear sorption, with results closely matching those from [10] and [24].
• For the turbulent flow model $ \partial_t u = \partial_x^2 u^{3/2} - (u^{3/2} - u^{1/2}) $, the numerical and analytical interfaces are visually indistinguishable, and the error remains low over time.
• The method accurately simulates the foam drainage model $ \partial_t u = \partial_x^2 u^{3/2} + \partial_x(u^2) $ and the viscous liquid advancing model $ \partial_t u = \partial_x^2 u^4 + \partial_x(u^3) $, reproducing expected physical behaviors such as symmetric spreading and front propagation.
• Comparisons with operator-splitting results from [24] show excellent agreement in both $ D=0.1 $ and $ D=0.001 $ cases, confirming robustness across different diffusion coefficients.