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A new class of energy dissipative, mass conserving and positivity/bound-preserving schemes for Keller-Segel equations

Zexiong Fang, Qing Cheng

Abstract

In this paper, we improve the original Lagrange multiplier approach \cite{ChSh22,ChSh_II22} and introduce a new energy correction approach to construct a class of robust, positivity/bound-preserving, mass conserving and energy dissipative schemes for Keller-Segel equations which only need to solve several linear Poisson like equations. To be more specific, we use a predictor-corrector approach to construct a class of positivity/bound-preserving and mass conserving schemes which can be implemented with negligible cost. Then a energy correction step is introduced to construct schemes which are also energy dissipative, in addition to positivity/bound-preserving and mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization to achieve high-order accuracy in time. We show stability results for mass-conservative, positivity/bound-preserving and energy dissipative schemes for two different Keller-Segel systems. A error analysis is presented for a second-order, bound-preserving, mass-conserving and energy dissipative scheme for the second-type of Keller-Segel equations. Ample numerical experiments are shown to validate the stability and accuracy of our approach.

A new class of energy dissipative, mass conserving and positivity/bound-preserving schemes for Keller-Segel equations

Abstract

In this paper, we improve the original Lagrange multiplier approach \cite{ChSh22,ChSh_II22} and introduce a new energy correction approach to construct a class of robust, positivity/bound-preserving, mass conserving and energy dissipative schemes for Keller-Segel equations which only need to solve several linear Poisson like equations. To be more specific, we use a predictor-corrector approach to construct a class of positivity/bound-preserving and mass conserving schemes which can be implemented with negligible cost. Then a energy correction step is introduced to construct schemes which are also energy dissipative, in addition to positivity/bound-preserving and mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization to achieve high-order accuracy in time. We show stability results for mass-conservative, positivity/bound-preserving and energy dissipative schemes for two different Keller-Segel systems. A error analysis is presented for a second-order, bound-preserving, mass-conserving and energy dissipative scheme for the second-type of Keller-Segel equations. Ample numerical experiments are shown to validate the stability and accuracy of our approach.

Paper Structure

This paper contains 20 sections, 13 theorems, 205 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Structure-preserving schemes
  3. Spatial discretization
  4. Energy correction approach based on CN
  5. Energy correction approach based on BDFk
  6. Bound-preserving schemes
  7. Mass conserving and bound-preserving schemes with BDFk
  8. Bound-preserving scheme based on CN
  9. Stability results
  10. Stability of positivity preserving schemes
  11. Stability of Crank-Nicolson scheme
  12. Stability of bound-preserving schemes
  13. Stability of Crank-Nicolson scheme
  14. Error estimate
  15. Numerical simulations
  16. ...and 5 more sections

Key Result

Lemma 2.1

\newlabelpos:lemma0 For the second-order scheme en:positivity:lag:1-en:positivity:lag:7, Lagrange multipliers $\lambda_h^{n+1}$ and $\xi_h^{n+1}$ satisfy

Figures (7)

  • Figure 1: Accuracy test: The $L^{\infty}$ errors of density $\rho(\bm{x},t)$ and concentration $c(\bm{x},t)$ at $t=0.01$ for the first type Keller-Segel equations \ref{['de:keller:1']}-\ref{['de:keller:3']} computed by positivity schemes \ref{['high:positivity:lag:1']}-\ref{['high:positivity:lag:4']} with $k=1,2$.
  • Figure 2: Accuracy test: The $L^{\infty}$ errors of density $\rho(\bm{x},t)$ and concentration $c(\bm{x},t)$ at $t=0.01$ for the second type Keller-Segel equations \ref{['keller:1']}-\ref{['keller:3']} computed by bound-preserving schemes \ref{['keller:lag:1']}-\ref{['keller:lag:6']} with $k=1,2$.
  • Figure 3: Numerical solutions $\rho(\bm{x},t)$ and Lagrange multiplier $\lambda(\bm{x},t)$ at $t=0,005,0.02$ computed by positivity preserving scheme \ref{['high:positivity:lag:1']}-\ref{['high:positivity:lag:4']} with $k=2$ and time step $\delta t=10^{-4}$.
  • Figure 4: (a)-(d): Concentration $c$ and density $\rho$ computed by second-order bound-preserving scheme \ref{['high:positivity:lag:1']}-\ref{['high:positivity:lag:4']} with time step $\delta t=10^{-4}$. (e) Lagrange multiplier $\lambda$. (f): Maximum and minimum value of $\rho$.
  • Figure 5: Numerical solution $\rho$ at $t=0,0.005,0.01,0.02$ computed by positivity preserving scheme \ref{['high:positivity:lag:1']}-\ref{['high:positivity:lag:4']} with $k=2$ and time step $\delta t=10^{-4}$.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Lemma 2.1
  • Proof 1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • Proof 2
  • Lemma 2.3
  • Lemma 2.4
  • Proof 3
  • Theorem 2.5
  • ...and 17 more