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Structure-preserving scheme for fractional nonlinear diffusion equations

Hélène Hivert, Florian Salin

Abstract

In this paper, we introduce and analyze a numerical scheme for solving the Cauchy-Dirichlet problem associated with fractional nonlinear diffusion equations. These equations generalize the porous medium equation and the fast diffusion equation by incorporating a fractional diffusion term. We provide a rigorous analysis showing that the discretization preserves main properties of the continuous equations, including algebraic decay in the fractional porous medium case and the extinction phenomenon in the fractional fast diffusion case. The study is supported by extensive numerical simulations. In addition, we propose a novel method for accurately computing the extinction time for the fractional fast diffusion equation and illustrate numerically the convergence of rescaled solutions towards asymptotic profiles near the extinction time.

Structure-preserving scheme for fractional nonlinear diffusion equations

Abstract

In this paper, we introduce and analyze a numerical scheme for solving the Cauchy-Dirichlet problem associated with fractional nonlinear diffusion equations. These equations generalize the porous medium equation and the fast diffusion equation by incorporating a fractional diffusion term. We provide a rigorous analysis showing that the discretization preserves main properties of the continuous equations, including algebraic decay in the fractional porous medium case and the extinction phenomenon in the fractional fast diffusion case. The study is supported by extensive numerical simulations. In addition, we propose a novel method for accurately computing the extinction time for the fractional fast diffusion equation and illustrate numerically the convergence of rescaled solutions towards asymptotic profiles near the extinction time.
Paper Structure (16 sections, 14 theorems, 189 equations, 7 figures)

This paper contains 16 sections, 14 theorems, 189 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Presentation of the schemes and main results
  3. Presentation of the schemes and notations
  4. Well-posedness of the schemes and convergence
  5. Decay estimates of the schemes
  6. Discrete Functional Analysis
  7. Proof of well-posedness of the schemes and convergence results
  8. Proof of energy decay estimates
  9. Numerical results
  10. Implementation
  11. Energy decay and Rayleigh quotient
  12. Convergence of the scheme
  13. Numerical computation of the extinction time
  14. Convergence to asymptotic profiles in the fast diffusion case
  15. The explicit scheme
  16. ...and 1 more sections

Key Result

Proposition 1

There exists positive constants $b_\theta$ and $B_\theta$ depending only on $\theta\in(0,1)$ such that

Figures (7)

  • Figure 1: Decay of $\|\mathbf{u}^n\|_{\ell^q_h(\mathbb{R})}$ for $L=1, \Delta t=0.001, h=0.05, \theta=0.25$ and $u^0$ given by \ref{['Eqn:DataIni']}.
  • Figure 2: Decay of $\|\mathbf{u}^n\|_{\mathcal{X}^\theta_h(I)}^2/\|\mathbf{u}^n\|_{\ell^q_h(\mathbb{R})}^2$ for $L=1, \Delta t=0.001, h=0.05, \theta=0.25$ and $u^0$ given by \ref{['Eqn:DataIni']}.
  • Figure 3: $\max_{n} \|u^{n}-u^{n}_{\Delta t,h}\|_{\ell^q_h(I)}$ for $T=2, h=0.05, L=1, \Delta t_{\rm ref}=5e-05$, and $u^0$ given by \ref{['Eqn:DataIni']}.
  • Figure 4: $\max_{n} \|u^{n}-u^{n}_{\Delta t,h}\|_{\ell^q_h(I)}$ for $L=1, \Delta t =0.01, T=2, h_{\rm ref}=2^{-11}$ and $u^0$ given by \ref{['Eqn:DataIni']}.
  • Figure 5: Convergence of the extinction time computed with the scheme described in subsection \ref{['Subsec:ExtinctionTime']}, for $q=2.4,\,L=1,\,\epsilon=0.5\cdot 10^{-8}$. (A): Convergence in $\Delta s$ with $h=0.1$ and reference extinction time computed with $\Delta s_{\rm ref}=5\cdot10^{-4}$. (B): Convergence in $h$ with $\Delta s=0.1$ and reference extinction time computed with $h_{\rm ref}=1\cdot 10^{-3}$.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Proposition 1
  • Remark 2
  • Theorem 3: Error estimate
  • Definition 4
  • Lemma 5
  • Proposition 6: Existence and stability for \ref{['Eqn:FD']}
  • Proposition 7: Existence and stability for \ref{['Eqn:SD']}
  • Proposition 8: Convergence of \ref{['Eqn:FD']}
  • Proposition 9: Decay estimates for \ref{['Eqn:SD']}
  • Proposition 10: Decay estimates for \ref{['Eqn:FD']}
  • ...and 8 more