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

## TL;DR

This paper develops and analyzes physical-constraints-preserving Lagrangian finite volume schemes for 1D and 2D special relativistic hydrodynamics, ensuring positivity of density and pressure, and bounded velocities, with high accuracy and effectiveness demonstrated through numerical experiments.

## Contribution

It introduces PCP Lagrangian schemes with proven positivity and boundedness properties for special relativistic hydrodynamics, including high-order accuracy techniques.

## Key findings

- Proved PCP property for first-order schemes with HLLC solver.
- Developed high-order schemes using SSP time discretization and WENO reconstruction.
- Numerical experiments confirm accuracy and robustness in complex RHD scenarios.

## 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.