This paper has a dual purpose: it presents a multidomain Ohebyshev method for the solution of the two-dimensional reactive compressible Navier-Stokes equations and it reports the results of the application of this code to the numerical simulations of high Mach number reactive flows in recessed cavity. The computational method utilizes newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations. The results of the simulations are relevant to recessed cavity flameholders.