# A direct primitive variable recovery scheme for hyperbolic conservative equations: the case of relativistic hydrodynamics

## TL;DR

This paper introduces a Primitive Variable Recovery Scheme (PVRS) for hyperbolic conservative equations, especially in relativistic hydrodynamics, which directly computes primitive variables avoiding complex algebraic solutions.

## Contribution

The PVRS method generalizes existing techniques by directly recovering primitive variables through the chain rule, simplifying computations in relativistic hydrodynamics.

## Key findings

- Demonstrates convergence on shock-tube problems
- Provides graphical error visualization
- Avoids algebraic polynomial solutions for primitive variables

## Abstract

In this article we develop a Primitive Variable Recovery Scheme (PVRS) to solve any system of coupled differential conservative equations. This method obtains directly the primitive variables applying the chain rule to the time term of the conservative equations. With this, a traditional finite volume method for the flux is applied in order avoid violation of both, the entropy and "Rankine-Hugoniot" jump conditions. The time evolution is then computed using a forward finite difference scheme. This numerical technique evades the recovery of the primitive vector by solving an algebraic system of equations as it is often used and so, it generalises standard techniques to solve these kind of coupled systems. The article is presented bearing in mind special relativistic hydrodynamic numerical schemes with an added pedagogical view in the appendix section in order to easily comprehend the PVRS. We present the convergence of the method for standard shock-tube problems of special relativistic hydrodynamics and a graphical visualisation of the errors using the fluctuations of the numerical values with respect to exact analytic solutions. The PVRS circumvents the sometimes arduous computation that arises from standard numerical methods techniques, which obtain the desired primitive vector solution through an algebraic polynomial of the charges.