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SAV-based entropy-dissipative schemes for a class of kinetic equations

Shiheng Zhang, Jie Shen, Jingwei Hu

Abstract

We introduce novel entropy-dissipative numerical schemes for a class of kinetic equations, leveraging the recently introduced scalar auxiliary variable (SAV) approach. Both first and second order schemes are constructed. Since the positivity of the solution is closely related to entropy, we also propose positivity-preserving versions of these schemes to ensure robustness, which include a scheme specially designed for the Boltzmann equation and a more general scheme using Lagrange multipliers. The accuracy and provable entropy-dissipation properties of the proposed schemes are validated for both the Boltzmann equation and the Landau equation through extensive numerical examples.

SAV-based entropy-dissipative schemes for a class of kinetic equations

Abstract

We introduce novel entropy-dissipative numerical schemes for a class of kinetic equations, leveraging the recently introduced scalar auxiliary variable (SAV) approach. Both first and second order schemes are constructed. Since the positivity of the solution is closely related to entropy, we also propose positivity-preserving versions of these schemes to ensure robustness, which include a scheme specially designed for the Boltzmann equation and a more general scheme using Lagrange multipliers. The accuracy and provable entropy-dissipation properties of the proposed schemes are validated for both the Boltzmann equation and the Landau equation through extensive numerical examples.
Paper Structure (17 sections, 5 theorems, 61 equations, 6 figures, 1 table)

This paper contains 17 sections, 5 theorems, 61 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Entropy-dissipative SAV schemes
  3. A first order scheme
  4. A second order scheme
  5. A positivity-preserving and entropy-dissipative SAV scheme for the Boltzmann equation
  6. Positivity-preserving schemes for general kinetic equations
  7. Positivity-preserving schemes without mass conservation
  8. A first order scheme
  9. A second order scheme
  10. Positivity-preserving schemes with mass conservation
  11. Numerical examples
  12. Test case 1: BKW solution
  13. Convergence tests
  14. Entropy evolution
  15. Positivity-preserving SAV scheme for the Boltzmann equation
  16. ...and 2 more sections

Key Result

Theorem 2.1

\newlabelThm-SAV-1st0 The scheme sav-landau-1-sav-landau-2, SAV-1st, satisfies the following properties: for all time steps $n\geq 0$ and step size $\Delta t>0$,

Figures (6)

  • Figure 1: Convergence tests of SAV-1st, SAV-2nd, SAV-1st-LM, and SAV-2nd-LM for the Boltzmann equation and Landau equation.
  • Figure 2: Solution profile and entropy evolution for the Boltzmann equation with different time step sizes. Top 4 figures: SAV-1st-LM. Bottom 4 figures: SAV-2nd-LM.
  • Figure 3: Solution profile and entropy evolution for the Landau equation with different time step sizes. Top 4 figures: SAV-1st-LM. Bottom 4 figures: SAV-2nd-LM.
  • Figure 4: Convergence test and entropy evolution for SAV-1st-P-B.
  • Figure 5: Solutions computed using SAV-2nd-LM for the Boltzmann equation (with $\Delta t=0.01$) and Landau equation (with $\Delta t=0.002$).
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • Proof 1
  • Theorem 2.2
  • Proof 2
  • Remark 2.3
  • Theorem 3.1
  • Proof 3
  • Theorem 4.1
  • Proof 4
  • Remark 4.2
  • ...and 3 more