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Fractional-step High-order and Bound-preserving Method for Convection Diffusion Equations

Baolin Kuang, Hongfei Fu, Shusen Xie

Abstract

In this paper, we derive two bound-preserving and mass-conserving schemes based on the fractional-step method and high-order compact (HOC) finite difference method for nonlinear convection-dominated diffusion equations. We split the one-dimensional equation into three stages, and employ appropriate temporal and spatial discrete schemes respectively. We show that our scheme is weakly monotonic and that the bound-preserving property can be achieved using the bound-preserving limiter under some mild step constraints. By employing the alternating direction implicit (ADI) method, we extend the scheme to two-dimensional problems, further reducing computational cost. We also provide various numerical experiments to verify our theoretical results.

Fractional-step High-order and Bound-preserving Method for Convection Diffusion Equations

Abstract

In this paper, we derive two bound-preserving and mass-conserving schemes based on the fractional-step method and high-order compact (HOC) finite difference method for nonlinear convection-dominated diffusion equations. We split the one-dimensional equation into three stages, and employ appropriate temporal and spatial discrete schemes respectively. We show that our scheme is weakly monotonic and that the bound-preserving property can be achieved using the bound-preserving limiter under some mild step constraints. By employing the alternating direction implicit (ADI) method, we extend the scheme to two-dimensional problems, further reducing computational cost. We also provide various numerical experiments to verify our theoretical results.
Paper Structure (21 sections, 17 theorems, 59 equations, 21 figures, 5 tables, 3 algorithms)

This paper contains 21 sections, 17 theorems, 59 equations, 21 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. One-dimensional problems
  3. Strang splitting temporal discretization
  4. HOC spatial discretization
  5. Discrete mass conservation
  6. Bound-preserving analysis
  7. Extension to two-dimensional problems
  8. Strang splitting time discretization
  9. HOC spatial discretization
  10. Discrete mass conservation
  11. Bound-preserving analysis
  12. Numerical tests
  13. One-dimensional problems
  14. Example 1
  15. Example 2
  16. ...and 6 more sections

Key Result

lemma thmcounterlemma

For any grid function $v\in \mathcal{V}_h$, we have

Figures (21)

  • Figure 1: Numerical solutions and the evoution of $Mass_{err}$ without/with the BP limiter at different time.
  • Figure 2: Comparisons of numerical solutions obtained without/with TVB or/and BP limiter.
  • Figure 3: Comparisons of $min(U,0)$ obtained without/with TVB or/and BP limiter.
  • Figure 4: Comparisons of numerical solutions obtained without/with BP limiter under $N =600, \tau= \frac{h}{3 \max \limits_{u}\left|f^{\prime}(u)\right|}$ and $\nu= 5 \times 10^{-4}$.
  • Figure 5: Numerical solutions at different time with $\nu = 10^{-3 },N =500,\tau=\frac{h}{2 \max{|f'|} }$.
  • ...and 16 more figures

Theorems & Definitions (34)

  • remark thmcounterremark
  • remark thmcounterremark
  • lemma thmcounterlemma
  • theorem 1
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • ...and 24 more