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Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

Shumo Cui, Alexander Kurganov, Kailiang Wu

Abstract

Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable or even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.

Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

Abstract

Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable or even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.
Paper Structure (18 sections, 15 theorems, 71 equations, 8 figures, 1 table)

This paper contains 18 sections, 15 theorems, 71 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. One-Dimensional BPCU Schemes
  3. CU Schemes: a Brief Overview
  4. BP Framework for CU Schemes
  5. BP Condition #1: $u^\pm_{j+\frac{1}{2}}\in{\mathcal{G}}$
  6. BP Condition #2: $u^*_{j+\frac{1}{2}}\in{\mathcal{G}}$
  7. BP Condition #3: $u^*_j\in{\mathcal{G}}$
  8. BP Condition #4: $u^{*,\pm}_{j+\frac{1}{2}}\in{\mathcal{G}}$
  9. Constructing BPCU Schemes
  10. BPCU schemes for 1-D Euler Equations of Gas Dynamics
  11. Enforcing the BP Condition #1
  12. The BP Conditions #2 and #3
  13. Enforcing the BP Condition #4
  14. Constructing BPCU Schemes
  15. Two-Dimensional BPCU Schemes
  16. ...and 3 more sections

Key Result

Theorem 2.3

The CU scheme 2--8 admits the following convex decomposition: under the CFL condition

Figures (8)

  • Figure 1: A possible configuration of $\, \hbox{${\bm u}$}$, ${\bm u}^\pm$, and $\bm s^\pm_\varepsilon$ (not to scale).
  • Figure 2: Flow chart of BPCU schemes from $\, \hbox{${\bm u}$}^{\,n}_j$ to $\, \hbox{${\bm u}$}^{\,n+1}_j$. To guarantee the BP property, the four formulae (blocks in yellow) may require careful redesign/verification and may differ from the counterparts of the original CU schemes.
  • Figure 3: \ref{['ex42']}: Density ($\rho$) computed by the BPCU scheme and Schemes 2 and 3 (left) and zoom at the smooth part of $\rho$ (right).
  • Figure 4: \ref{['ex43']}: Density ($\rho$) and pressure ($p$) computed by the BPCU scheme.
  • Figure 5: \ref{['ex44']}: Solution ($\ln\rho$ (top row) and $\ln p$ (bottom row)) computed by the BPCU scheme at times $t=0.05$ (left column) and 0.07 (right column).
  • ...and 3 more figures

Theorems & Definitions (32)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.1
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 22 more