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Bound-Preserving WENO Schemes for Temple-class systems

Wei Chen, Shumo Cui, Kailiang Wu, Tao Xiong, Baoyue Yu

TL;DR

The paper tackles the challenge of accurately simulating Temple-class hyperbolic systems under non-convex invariant-domain constraints by introducing a moving-mesh bound-preserving framework. It establishes that BP is generally impossible on fixed meshes for these systems, and shows that a moving mesh with grid speed matching the fluid velocity enables stable, conservative BP schemes. The authors develop both first-order and high-order BP schemes, the latter incorporating a novel parameterized flux limiter to maintain invariant-domain constraints while achieving high accuracy via WENO and RK3 time stepping. Numerical experiments on Temple-class systems and ARZ traffic networks demonstrate robust bound preservation, positivity of $\phi$, and avoidance of velocity overshoot, with local invariant-domain estimates outperforming global ones. The approach offers a versatile, practical toolkit for reliable simulations across traffic, sedimentation, and other Temple-class applications.

Abstract

This paper explores numerical schemes for Temple-class systems, which are integral to various applications including one-dimensional two-phase flow, elasticity, traffic flow, and sedimentation. Temple-class systems are characterized by conservative equations, with different pressure function expressions leading to specific models such as the Aw-Rascle-Zhang (ARZ) traffic model and the sedimentation model. Our work extends existing studies by introducing a moving mesh approach to address the challenges of preserving non-convex invariant domains, a common issue in the numerical simulation of such systems. Our study outlines a novel bound-preserving (BP) and conservative numerical scheme, designed specifically for non-convex sets in Temple-class systems, which is critical for avoiding non-physical solutions and ensuring robustness in simulations. We develop both local and global BP methods based on finite difference schemes, with numerical experiments demonstrating the effectiveness and reliability of our methods. Furthermore, a parameterized flux limiter is introduced to restrict high-order fluxes and maintain bound preservation. This innovation marks the first time such a parameterized approach has been applied to non-convex sets, offering significant improvements over traditional methods. The findings presented extend beyond theoretical implications, as they are applicable to general Temple-class systems and can be tailored to ARZ traffic flow networks, highlighting the versatility and broad applicability of our approach. The paper contributes significantly to the field by providing a comprehensive method that maintains the physical and mathematical constrains of Temple-class systems.

Bound-Preserving WENO Schemes for Temple-class systems

TL;DR

The paper tackles the challenge of accurately simulating Temple-class hyperbolic systems under non-convex invariant-domain constraints by introducing a moving-mesh bound-preserving framework. It establishes that BP is generally impossible on fixed meshes for these systems, and shows that a moving mesh with grid speed matching the fluid velocity enables stable, conservative BP schemes. The authors develop both first-order and high-order BP schemes, the latter incorporating a novel parameterized flux limiter to maintain invariant-domain constraints while achieving high accuracy via WENO and RK3 time stepping. Numerical experiments on Temple-class systems and ARZ traffic networks demonstrate robust bound preservation, positivity of , and avoidance of velocity overshoot, with local invariant-domain estimates outperforming global ones. The approach offers a versatile, practical toolkit for reliable simulations across traffic, sedimentation, and other Temple-class applications.

Abstract

This paper explores numerical schemes for Temple-class systems, which are integral to various applications including one-dimensional two-phase flow, elasticity, traffic flow, and sedimentation. Temple-class systems are characterized by conservative equations, with different pressure function expressions leading to specific models such as the Aw-Rascle-Zhang (ARZ) traffic model and the sedimentation model. Our work extends existing studies by introducing a moving mesh approach to address the challenges of preserving non-convex invariant domains, a common issue in the numerical simulation of such systems. Our study outlines a novel bound-preserving (BP) and conservative numerical scheme, designed specifically for non-convex sets in Temple-class systems, which is critical for avoiding non-physical solutions and ensuring robustness in simulations. We develop both local and global BP methods based on finite difference schemes, with numerical experiments demonstrating the effectiveness and reliability of our methods. Furthermore, a parameterized flux limiter is introduced to restrict high-order fluxes and maintain bound preservation. This innovation marks the first time such a parameterized approach has been applied to non-convex sets, offering significant improvements over traditional methods. The findings presented extend beyond theoretical implications, as they are applicable to general Temple-class systems and can be tailored to ARZ traffic flow networks, highlighting the versatility and broad applicability of our approach. The paper contributes significantly to the field by providing a comprehensive method that maintains the physical and mathematical constrains of Temple-class systems.

Paper Structure

This paper contains 13 sections, 5 theorems, 68 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Temple-class system
  3. Related works
  4. Contributions
  5. The feasibility of bound-preserving schemes
  6. First-order BP scheme
  7. High-order BP scheme
  8. Approximate invariant domains
  9. Parametrized BP flux limiter
  10. Numerical Examples
  11. Numerical experiments for Temple-class systems
  12. Numerical experiments for ARZ model on road networks
  13. Conclusion

Key Result

Theorem 2.1

Consider a Temple-class system satisfying Assumption assum1 with the Riemann initial conditions: where $v_L = v_R = v^{\dag}$ and $\phi > 0$. For fixed computational mesh, no conservative and consistent scheme under the CFL condition $0 < \Delta t_n v^{\dag} / \Delta x \leq 1$ will produce an approximate solution contained in $\mathcal{I}^{\rm G} \cap \{\phi > 0\}$, and the upper bound $v_{\rm

Figures (14)

  • Figure 1: The $y (v, \phi)$ curves in Theorem \ref{['contradiction']}.
  • Figure 2: Limit $(v_{\min})_j^{n+1} \leq v_j^{n+1} \leq (v_{\max})_j^{n+1}$.
  • Figure 3: Example \ref{['exam2']}, the numerical resluts obtained by using the BP scheme and NonBP scheme, $t = 1$, and $\Delta \xi = 0.008$.
  • Figure 4: Example \ref{['exam3']}, Low: low-order BP scheme with $N_\xi = 500$; BP: high-order BP scheme with $N_\xi = 500$; Ref: low-order BP scheme with $N_\xi = 40000$.
  • Figure 5: Example \ref{['exam4']}, $t = 1$, and $N_\xi = 500$. Top: Test T1; Bottom: Test T2.
  • ...and 9 more figures

Theorems & Definitions (24)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 14 more