The aim of this article is to thoroughly identify the spurious mode that jeopardizes the convergence of usual upwind numerical schemes for compressible flows when the Mach number goes to 0. We show that this spurious mode is the long time limit of a wave system whose properties and discretization depend on the scheme used for the compressible system. Once this spurious mode is identified, a filtering method is developed for removing it from the solution of stationary low Mach number compressible flow. Numerical results confirm that at the price of the computation of a long time solution of the wave system, the accuracy of an inaccurate solution of a low Mach number compressible flow can be greatly improved by this filtering method