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Two difficulties are always encountered in the simulation of variable density low Mach number flows by the primitive variable methods: (1) the decoupling of pressure and velocity caused by the extremely small velocity; (2) the spurious oscillations, nonsmooth solutions and sharp resolution caused by discontinuities, making that the flow cannot be successfully simulated by low-ordered schemes. In this paper, a pressure-based compressible flow solver with a scheme of high-resolution is developed to overcome these difficulties. In the present work, we adopt pressure correction algorithm to overcome the decoupling and the MUSCL scheme to predict the discontinuities at low Mach number. We applied the proposed compressible flow solver to simulate the compressible flows in a planar nozzle, arc bumps and the lid-driven cavity and found that the numerical results are in good agreement with those reported in the previous works, indicating that the numerical algorithm developed in this work is a reliable and accurate tool for studying thermal variable density low Mach number flows.