# Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations

## TL;DR

This paper introduces second-order accurate genuine BGK schemes based on the analytical solution of the Anderson-Witting model for simulating ultra-relativistic flows, including viscous effects, with improved accuracy and resolution.

## Contribution

It presents the first derivation of genuine BGK schemes from the Anderson-Witting model for ultra-relativistic flows, incorporating particle collisions explicitly.

## Key findings

- Accurate and stable for ultra-relativistic inviscid and viscous flows.
- Higher resolution at contact discontinuities compared to existing schemes.
- Validated through multiple 1D and 2D numerical experiments.

## Abstract

This paper presents second-order accurate genuine BGK (Bhatnagar-Gross-Krook) schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present genuine BGK schemes are derived from the analytical solution of the Anderson-Witting model, which is given for the first time and includes the "genuine" particle collisions in the gas transport process. The BGK schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed BGK schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.