Hypersonic flows regard a special class of flows where dire conditions for spacecraft may be found. Simulation of these flows requires taking chemistry and thermal nonequilibrium into account as to acquire the correct wall quantities for the vehicles. Flux reconstruction -- a higher order method -- enables fluid simulations with a coarser grid than finite volume, as well as with easier shock discretisation for the practitioner. Computational Object-Oriented Libraries for Fluid Dynamics (COOLFluiD) allows the simulation of hypersonic flows with flux reconstruction, but not yet with thermochemical nonequilibrium due to positivity issues encountered. A survey of the literature for positivity methods in higher order methods, together with a presentation of flux reconstruction, the chemical model used and COOLFluiD were given. An entropy based exponential filter and a Fejér filter were implemented in COOLFluiD based on a pre-existing framework of filter based positivity and tested against the previous positivity method in place. The exponential filter was found to be equivalent for a 2D simulation with the Euler equations to the previous method while the Féjer filter behaved in a more destabilising manner. The entropy based exponential filter was implemented for cases with multiple species and multiple temperatures, with modularity for species sets. The derivation of entropy for this case was given. A new positivity method based on least squares optimisation was developed to target density positivity issues where the filter fails. The case was tested against two cases from literature in P0, ensuring positivity and validated against literature. The cases were found not to be yet fully spatially resolved, as expected in P0 simulations. P1 simulations were tested for the same cases and found to have issues with the time integration method and the artificial viscosity. An inspection of the code base was executed, revealing that the artificial viscosity was not fully implemented for the cases tested and that the current base implementation of backwards Euler cannot be used for these cases, requiring being solved through the already implemented Newton iterations method. The artificial viscosity was fully implemented for the tested cases. The P1 simulations were retested. The artificial viscosity was found to be active and to diffuse the shock wave. No positivity failures of the method were found. A stringent CFL condition was found to exist, and possibly linked to the stiffness of the equations. The exponential filter was found to lead to a growth of the temperatures just upstream of the shock wave, where the filter is active. This behaviour was linked to a possible interaction with the highly nonlinear model or with the artificial viscosity. While positivity was found to work, further issues in testing it are still found and expected to be part of future work. Possible extensions or alterations to the code were found to possibly be required in order to allow further testing. Thus, positivity was found not to be the only issue with running thermochemical nonequilibrium simulations, being only part of the problem, requiring further research in order to fully test it in the context of COOLFluiD. Future work was defined to focus on studying the exponential filter interactions, on enabling future tests, on expanding on these testes based on presented literature, and on expanding on the model used to include more robust but complicated methods.