A new computation procedure is developed for solving time-accurate, chemically reacting flows over a wide range of Mach numbers. The algorithm is based on scaling the pressure terms in the momentum equations and preconditioning the conservation equations to circumvent numerical difficulties at low Mach numbers. The resulting equations are solved using the lower-upper (LU) factorization method in a fully-coupled manner, with incorporation of a flux-differencing upwind TVD scheme to achieve high-order spatial accuracy. The transient behavior of the modeled system is preserved through implementation of the dual time-stepping integration technique. The capabilities of the scheme are illustrated by applying it to selected problems, including one-dimensional nozzle flows, two-dimensional channel flows, rocket-motor internal flows, and acoustic waves in a porous chamber with surface transpiration.