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Novel Local Characteristic Decomposition Based Path-Conservative Central-Upwind Schemes

Shaoshuai Chu, Michael Herty, Alexander Kurganov

TL;DR

This work develops local characteristic decomposition based path-conservative central-upwind (LCD-PCCU) schemes for nonconservative hyperbolic systems of balance laws, achieving well-balancedness through flux globalization that embeds source terms into global fluxes and using path-conservative techniques for nonconservative products. It first introduces a 1-D LCD-CU method, then extends to 1-D and 2-D flux globalization based LCD-PCCU schemes that reconstruct equilibrium variables to preserve discrete steady states, with eigen-structure driven fluxes and desingularization to maintain robustness. The methods are tested on gamma-based compressible multifluids and thermal rotating shallow water equations, showing sharper interface resolution and improved accuracy over traditional PCCU schemes across multiple 1-D and 2-D scenarios. The results demonstrate the practical value of LCD-PCCU in simulating complex multi-fluid and geophysical flows with strong nonconservative effects, while maintaining stability and efficiency.

Abstract

We introduce local characteristic decomposition based path-conservative central-upwind schemes for (nonconservative) hyperbolic systems of balance laws. The proposed schemes are made to be well-balanced via a flux globalization approach, in which source terms are incorporated into the fluxes: This helps to enforce the well-balanced property when the resulting quasi-conservative system is solved using the local characteristic decomposition based central-upwind scheme recently introduced in [{\sc A. Chertock, S. Chu, M. Herty, A. Kurganov, and M. Lukáčová-Medvi{\softd}ová}, J. Comput. Phys., 473 (2023), Paper No. 111718]. Nonconservative product terms are also incorporated into the global fluxes using a path-conservative technique. We illustrate the performance of the developed schemes by applying them to one- and two-dimensional compressible multifluid systems and thermal rotating shallow water equations.

Novel Local Characteristic Decomposition Based Path-Conservative Central-Upwind Schemes

TL;DR

This work develops local characteristic decomposition based path-conservative central-upwind (LCD-PCCU) schemes for nonconservative hyperbolic systems of balance laws, achieving well-balancedness through flux globalization that embeds source terms into global fluxes and using path-conservative techniques for nonconservative products. It first introduces a 1-D LCD-CU method, then extends to 1-D and 2-D flux globalization based LCD-PCCU schemes that reconstruct equilibrium variables to preserve discrete steady states, with eigen-structure driven fluxes and desingularization to maintain robustness. The methods are tested on gamma-based compressible multifluids and thermal rotating shallow water equations, showing sharper interface resolution and improved accuracy over traditional PCCU schemes across multiple 1-D and 2-D scenarios. The results demonstrate the practical value of LCD-PCCU in simulating complex multi-fluid and geophysical flows with strong nonconservative effects, while maintaining stability and efficiency.

Abstract

We introduce local characteristic decomposition based path-conservative central-upwind schemes for (nonconservative) hyperbolic systems of balance laws. The proposed schemes are made to be well-balanced via a flux globalization approach, in which source terms are incorporated into the fluxes: This helps to enforce the well-balanced property when the resulting quasi-conservative system is solved using the local characteristic decomposition based central-upwind scheme recently introduced in [{\sc A. Chertock, S. Chu, M. Herty, A. Kurganov, and M. Lukáčová-Medvi{\softd}ová}, J. Comput. Phys., 473 (2023), Paper No. 111718]. Nonconservative product terms are also incorporated into the global fluxes using a path-conservative technique. We illustrate the performance of the developed schemes by applying them to one- and two-dimensional compressible multifluid systems and thermal rotating shallow water equations.
Paper Structure (17 sections, 63 equations, 17 figures)

This paper contains 17 sections, 63 equations, 17 figures.

Table of Contents

  1. Introduction
  2. 1-D LCD-CU Scheme
  3. 1-D Flux Globalization Based LCD-PCCU Scheme
  4. 2-D Flux Globalization Based LCD-PCCU Scheme
  5. Numerical Examples
  6. $\gamma$-Based Compressible Multifluid Equations
  7. One-Dimensional Examples
  8. Two-Dimensional Examples
  9. Thermal Rotating Shallow Water Equations
  10. One-Dimensional Examples
  11. Two-Dimensional Examples
  12. LCD Matrices for the $\gamma$-Based Compressible Multifluid Equations
  13. One-Dimensional Case
  14. Two-Dimensional Case
  15. LCD Matrices for the TRSW Equations
  16. ...and 2 more sections

Figures (17)

  • Figure 5.1: Example 1: Density $\rho$ computed by the PCCU and LCD-PCCU schemes (left) and zoom at $x\in[-0.55,-0.4]$ (right).
  • Figure 5.2: Example 2: Density $\rho$ computed by the PCCU and LCD-PCCU schemes (left) and zoom at $x\in[2.5,4.8]$ (right).
  • Figure 5.3: Initial setting for the 2-D numerical examples.
  • Figure 5.4: Example 3: Shock-helium bubble interaction by the PCCU (left column) and LCD-PCCU (right column) schemes at different times.
  • Figure 5.5: Example 4: Shock-R22 bubble interaction by the PCCU (left column) and LCD-PCCU (right column) schemes at different times.
  • ...and 12 more figures

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 4.1
  • Remark 5.1
  • Remark 5.2
  • Remark 5.3
  • Remark 5.4