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Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics

Dan Ling, Junming Duan, Huazhong Tang

TL;DR

The paper develops physical-constraints-preserving Lagrangian finite volume schemes for 1D and 2D special relativistic hydrodynamics using the HLLC Riemann solver. It first proves PCP for a first-order 1D scheme under suitable wave-speed estimates, ensuring density, pressure positivity, and subluminal velocity, then constructs high-order PCP schemes via WENO reconstructions, a scaling limiter, and SSP time discretization. The extension to 2D preserves PCP through edge-based HLLC fluxes on quadrilateral meshes, addressing extra couplings from tangential velocities; a second-order accurate 2D scheme is achieved with 2D WENO-like reconstructions and Gauss-Lobatto quadrature, accompanied by PCP limiting. Extensive 1D and 2D numerical experiments demonstrate that the PCP schemes remain robust and accurate in the presence of strong discontinuities, large Lorentz factors, and low-density/low-pressure regimes, validating both the theory and practical utility of the approach.

Abstract

This paper studies the physical-constraints-preserving (PCP) Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamic (RHD) equations. First, the PCP property (i.e. preserving the positivity of the rest-mass density and the pressure and the bound of the velocity) is proved for the first-order accurate Lagrangian scheme with the HLLC Riemann solver and forward Euler time discretization. The key is that the intermediate states in the HLLC Riemann solver are shown to be admissible or PCP when the HLLC wave speeds are estimated suitably. Then, the higher-order accurate schemes are proposed by using the high-order accurate strong stability preserving (SSP) time discretizations and the scaling PCP limiter as well as the WENO reconstruction. Finally, several one- and two-dimensional numerical experiments are conducted to demonstrate the accuracy and the effectiveness of the PCP Lagrangian schemes in solving the special RHD problems involving strong discontinuities, or large Lorentz factor, or low rest-mass density or low pressure etc.

Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics

TL;DR

The paper develops physical-constraints-preserving Lagrangian finite volume schemes for 1D and 2D special relativistic hydrodynamics using the HLLC Riemann solver. It first proves PCP for a first-order 1D scheme under suitable wave-speed estimates, ensuring density, pressure positivity, and subluminal velocity, then constructs high-order PCP schemes via WENO reconstructions, a scaling limiter, and SSP time discretization. The extension to 2D preserves PCP through edge-based HLLC fluxes on quadrilateral meshes, addressing extra couplings from tangential velocities; a second-order accurate 2D scheme is achieved with 2D WENO-like reconstructions and Gauss-Lobatto quadrature, accompanied by PCP limiting. Extensive 1D and 2D numerical experiments demonstrate that the PCP schemes remain robust and accurate in the presence of strong discontinuities, large Lorentz factors, and low-density/low-pressure regimes, validating both the theory and practical utility of the approach.

Abstract

This paper studies the physical-constraints-preserving (PCP) Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamic (RHD) equations. First, the PCP property (i.e. preserving the positivity of the rest-mass density and the pressure and the bound of the velocity) is proved for the first-order accurate Lagrangian scheme with the HLLC Riemann solver and forward Euler time discretization. The key is that the intermediate states in the HLLC Riemann solver are shown to be admissible or PCP when the HLLC wave speeds are estimated suitably. Then, the higher-order accurate schemes are proposed by using the high-order accurate strong stability preserving (SSP) time discretizations and the scaling PCP limiter as well as the WENO reconstruction. Finally, several one- and two-dimensional numerical experiments are conducted to demonstrate the accuracy and the effectiveness of the PCP Lagrangian schemes in solving the special RHD problems involving strong discontinuities, or large Lorentz factor, or low rest-mass density or low pressure etc.

Paper Structure

This paper contains 15 sections, 8 theorems, 167 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. One-dimensional PCP Lagrangian schemes
  3. First-order accurate scheme
  4. High-order accurate scheme
  5. Scaling PCP limiter
  6. High-order time discretization
  7. Two-dimensional PCP Lagrangian schemes
  8. First-order accurate scheme
  9. High-order accurate scheme
  10. Numerical results
  11. 1D case
  12. 2D case
  13. Conclusions
  14. Proof of Lemma \ref{['lem3']}
  15. Proof of Lemma \ref{['lem4']}

Key Result

Lemma 2.1

For the given $\hbox{\boldmath \small $U$}^{\pm}=(D^{\pm},m^{\pm},E^{\pm})^{T}$, if the wave speed $s^{\pm}$ are estimated in ws and $s^{\ast}$ is the speed of the contact discontinuity, then $A^{\pm}$ and $B^{\pm}$ defined in eq67 satisfy the following inequalities

Figures (13)

  • Figure 2.1: Structure of the exact or HLLC approximate solution of the Riemann problem \ref{['eq11']}.
  • Figure 2.2: Diagrammatic sketch of the neighboring Riemann problems.
  • Figure 3.1: A general quadrilateral cell.
  • Figure 3.2: The local coordinates and the stencils in the WENO reconstruction.
  • Figure 3.3: The quadrilateral in the $(x,y)$ plane mapped into the square in the $(\hat{x}, \hat{y})$ plane.
  • ...and 8 more figures

Theorems & Definitions (18)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • ...and 8 more