In this thesis, we discuss first and second order finite volume methods to solve the one dimensional compressible Navier-Stokes equations. We prove the first order finite volume method preserves positivity for the density and pressure. We carry out a sequence of numerical tests including the famous Shock tube problem, extreme Riemann double rarefaction wave case, etc. For those cases with very low density, our scheme performed well and the density and pressure remain positive throughout the domain. We further consider to extend the positivity preserving discussion to second order finite volume methods.