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A Comparative Study of Low-Dissipation Numerical Schemes for Hyperbolic Conservation Laws

Shaoshuai Chu, Michael Herty

Abstract

This work provides a comparative assessment of several low-dissipation numerical schemes for hyperbolic conservation laws, highlighting their performance relative to the classical Harten-Lax-van Leer (HLL) schemes. The schemes under consideration include the classical Harten-Lax-van Leer-Contact (HLLC), the recently proposed TV flux splitting, the low-dissipation Central-Upwind (LDCU), and the local characteristic decomposition-based Central-Upwind (LCDCU) schemes. These methods are extended to higher orders of accuracy, up to the fifth order, within both finite-volume and finite-difference frameworks. A series of numerical experiments for the one- and two-dimensional Euler equations of gas dynamics are performed to evaluate the accuracy, robustness, and computational efficiency of the studied schemes. The comparison highlights the trade-offs between resolution of contact and shear waves, robustness in the presence of shocks, and computational cost. The investigated low-dissipation schemes show comparable levels of numerical dissipation, with only subtle differences appearing in selected benchmark problems. The results provide practical guidance for selecting efficient low-dissipation solvers for the simulation of complex compressible flows.

A Comparative Study of Low-Dissipation Numerical Schemes for Hyperbolic Conservation Laws

Abstract

This work provides a comparative assessment of several low-dissipation numerical schemes for hyperbolic conservation laws, highlighting their performance relative to the classical Harten-Lax-van Leer (HLL) schemes. The schemes under consideration include the classical Harten-Lax-van Leer-Contact (HLLC), the recently proposed TV flux splitting, the low-dissipation Central-Upwind (LDCU), and the local characteristic decomposition-based Central-Upwind (LCDCU) schemes. These methods are extended to higher orders of accuracy, up to the fifth order, within both finite-volume and finite-difference frameworks. A series of numerical experiments for the one- and two-dimensional Euler equations of gas dynamics are performed to evaluate the accuracy, robustness, and computational efficiency of the studied schemes. The comparison highlights the trade-offs between resolution of contact and shear waves, robustness in the presence of shocks, and computational cost. The investigated low-dissipation schemes show comparable levels of numerical dissipation, with only subtle differences appearing in selected benchmark problems. The results provide practical guidance for selecting efficient low-dissipation solvers for the simulation of complex compressible flows.
Paper Structure (28 sections, 95 equations, 21 figures, 3 tables)

This paper contains 28 sections, 95 equations, 21 figures, 3 tables.

Table of Contents

  1. Introduction
  2. One-Dimensional Schemes
  3. One-Dimensional HLL Schemes
  4. One-Dimensional First-Order HLL Scheme
  5. One-Dimensional Second-Order HLL Scheme
  6. One-Dimensional Third-Order HLL Scheme
  7. One-Dimensional Fifth-Order HLL Scheme
  8. One-Dimensional HLLC Schemes
  9. One-Dimensional TV Splitting Schemes
  10. One-Dimensional LDCU Schemes
  11. One-Dimensional LCDCU Schemes
  12. Two-Dimensional Schemes
  13. Two-Dimensional HLL Schemes
  14. Two-Dimensional First-Order HLL Scheme
  15. Two-Dimensional Second-Order HLL Scheme
  16. ...and 13 more sections

Figures (21)

  • Figure 4.1: Example 2: Density $\rho$ computed by the 1-Order, 2-Order, 3-Order and 5-Order schemes (top row) and zoom at $[0.42, 0.62]$ (bottom row).
  • Figure 4.2: Example 3: Density $\rho$ computed by the 1-Order, 2-Order, 3-Order, and 5-Order schemes (top row) and zoom at $[2, 4.8]$ (bottom row).
  • Figure 4.3: Example 4: Density $\rho$ computed by the 1-Order, 2-Order, 3-Order, and 5-Order schemes (top row) and zoom at $[-0.85,0.25]$ (bottom row).
  • Figure 4.4: Example 5: Density $\rho$ computed by the 1-Order and 2-Order schemes (left column) and zoom at $[1.5,2.5]$ (middle column), and $[-2.8,-1.3]$ (right column).
  • Figure 4.5: Example 5: Density $\rho$ computed by the 3- and 5-Order schemes (left column) and zoom at $[1.5,2.5]$ (middle column), and $[-2.8,-1.3]$ (right column).
  • ...and 16 more figures

Theorems & Definitions (1)

  • Remark 4.1