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Robust time-discretisation and linearisation schemes for singular and degenerate evolution systems modelling biofilm growth

R. K. H. Smeets, K. Mitra, I. S. Pop, S. Sonner

TL;DR

The paper addresses numerical challenges in simulating degenerate quasilinear evolution systems that model biofilm growth by introducing a semi-implicit time discretisation that decouples a parabolic diffusion equation from a coupled second equation. It develops two robust linearisation schemes, the L-scheme and the M-scheme, within a split formulation that handles both degeneracy and possible singularities, and proves well-posedness and convergence of the time-discrete solutions to the continuous problem. Theoretical results show unconditional convergence of the schemes with contraction in the non-degenerate setting, and numerical experiments using finite elements demonstrate robustness and efficiency, with the M-scheme often outperforming a regularized Newton approach in practice. The methods are applicable to biofilm models and related porous-media and reactive-diffusion systems, offering structure-preserving, space-discretisation-agnostic tools for challenging degenerate problems and suggesting extensions to multi-substrate and cross-diffusion contexts.

Abstract

We propose and analyse numerical schemes for a system of quasilinear, degenerate evolution equations modelling biofilm growth as well as other processes such as flow through porous media and the spreading of wildfires. The first equation in the system is parabolic and exhibits degenerate and singular diffusion, while the second is either uniformly parabolic or an ordinary differential equation. First, we introduce a semi-implicit time discretisation that has the benefit of decoupling the equations. We prove the positivity, boundedness, and convergence of the time-discrete solutions to the time-continuous solution. Then, we introduce an iterative linearisation scheme to solve the resulting nonlinear time-discrete problems. Under weak assumptions on the time-step size, we prove that the scheme converges irrespective of the space discretisation and mesh. Moreover, if the problem is non-degenerate, the convergence becomes faster as the time-step size decreases. Finally, employing the finite element method for the spatial discretisation, we study the behaviour of the scheme, and compare its performance to other commonly used schemes. These tests confirm that the proposed scheme is robust and fast.

Robust time-discretisation and linearisation schemes for singular and degenerate evolution systems modelling biofilm growth

TL;DR

The paper addresses numerical challenges in simulating degenerate quasilinear evolution systems that model biofilm growth by introducing a semi-implicit time discretisation that decouples a parabolic diffusion equation from a coupled second equation. It develops two robust linearisation schemes, the L-scheme and the M-scheme, within a split formulation that handles both degeneracy and possible singularities, and proves well-posedness and convergence of the time-discrete solutions to the continuous problem. Theoretical results show unconditional convergence of the schemes with contraction in the non-degenerate setting, and numerical experiments using finite elements demonstrate robustness and efficiency, with the M-scheme often outperforming a regularized Newton approach in practice. The methods are applicable to biofilm models and related porous-media and reactive-diffusion systems, offering structure-preserving, space-discretisation-agnostic tools for challenging degenerate problems and suggesting extensions to multi-substrate and cross-diffusion contexts.

Abstract

We propose and analyse numerical schemes for a system of quasilinear, degenerate evolution equations modelling biofilm growth as well as other processes such as flow through porous media and the spreading of wildfires. The first equation in the system is parabolic and exhibits degenerate and singular diffusion, while the second is either uniformly parabolic or an ordinary differential equation. First, we introduce a semi-implicit time discretisation that has the benefit of decoupling the equations. We prove the positivity, boundedness, and convergence of the time-discrete solutions to the time-continuous solution. Then, we introduce an iterative linearisation scheme to solve the resulting nonlinear time-discrete problems. Under weak assumptions on the time-step size, we prove that the scheme converges irrespective of the space discretisation and mesh. Moreover, if the problem is non-degenerate, the convergence becomes faster as the time-step size decreases. Finally, employing the finite element method for the spatial discretisation, we study the behaviour of the scheme, and compare its performance to other commonly used schemes. These tests confirm that the proposed scheme is robust and fast.
Paper Structure (23 sections, 24 theorems, 143 equations, 7 figures, 1 algorithm)

This paper contains 23 sections, 24 theorems, 143 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Time discretisation
  4. Linearisation
  5. Preliminaries
  6. Functional setting and background
  7. Structural assumptions
  8. Weak formulation of the time continuous problem
  9. Time discretisation
  10. Proof of \ref{['thm: main time 1']}: well-posedness, positivity, and boundedness
  11. Existence and uniqueness
  12. Positivity and boundedness in $L^\infty(\Omega)$
  13. Proof of \ref{['theo:time_convergence']}: convergence of the time-discrete solutions
  14. Uniform boundedness of the interpolates in Bochner spaces
  15. Convergence to the time-continuous solution if $u_0\in H^1_0(\Omega)$
  16. ...and 8 more sections

Key Result

Theorem 1

For all sufficiently small time steps $\tau$, there exists a unique weak solution of eq: time-discrete. Moreover, the time-discrete solutions $u_n,v_n$ are positive and $u_n$ is bounded almost everywhere in $\Omega$. In particular, if $\Phi$ is singular, $u_n$ is bounded by a constant strictly less

Figures (7)

  • Figure 1: Error in \ref{['eq:Conv-H1']} against time step size $\tau$ for $h = 10^{-4}$, $m=4$, $\gamma = 1/3$, time $0.5 \leq t \leq 1$, $\text{tol} = 10^{-7}$, $d=1$, $\beta = 1$.
  • Figure 2: Average iterations required for solving \ref{['eq: mod PME']} in 1D for varying mesh size $h$ and time steps $\tau$, with $m=4$, $\gamma = 1/3$, for time $0.5 \leq t \leq 1.1$, $\text{tol} = 10^{-5}$.
  • Figure 3: Convergence rate $\alpha$ against time step size $\tau$ for $h = 10^{-4}$, $m=4$, $\gamma = 1/3$, $t = 0.5$, $d=1$.
  • Figure 4: Average iterations required for solving \ref{['eq: PDE-PDE ex num']} in 1D for varying mesh size $h$ and time steps $\tau$, with $m=4$, $\gamma = 1/4$, for time $0 \leq t \leq 1.2$, $\mu = 0$ and $\text{tol} = 10^{-5}$.
  • Figure 5: Convergence rate $\alpha$ against time step size $\tau$ for $h = 10^{-4}$, $m=4$, $\gamma = 1/4$.
  • ...and 2 more figures

Theorems & Definitions (60)

  • Theorem : Well-posedness, boundedness, and convergence of the time-discrete solutions
  • Theorem : Convergence of the L-scheme
  • Theorem : Convergence of the M-scheme
  • Remark 2.0.1: Validity of the assumptions \ref{['ass: 1']} - \ref{['ass: 3']}
  • Definition 2.1: Weak formulation
  • Remark 3.0.1: The decoupling of the equations
  • Theorem 3.1: Well-posedness and boundedness of the time-discrete solutions
  • Remark 3.1.1: Computable upper bound for $u_n$
  • Theorem 3.2: Convergence of the time-discrete solutions
  • Lemma 3.3: Well-posedness for \ref{['eq: time-discrete weak system']}
  • ...and 50 more