In [J. Comput. Phys. 192(1), pp.325-354 (2003)], we have developed a multi-domain spectral method with stable and conservative penalty interface conditions for the numerical simulation of supersonic reactive recessed cavity flows with homogeneous grid. In this work, the previously developed methodology is generalized to inhomogeneous grid to simulate the two dimensional supersonic injector-cavity system. Non-physical modes in the solution generated at the domain interfaces due to the spatial grid inhomogeneity are minimized using the new weighted multi-domain spectral penalty method. The proposed method yields accurate and stable solutions of the injector-cavity system which agree well with experiments qualitatively. Through the direct numerical simulation of the injector-cavity system using the weighted method, the geometric effect of the cavity wall on pressure fluctuations is investigated. It is shown that the recessed slanted cavity attenuates pressure fluctuations inside cavity enabling the cavity to act potentially as a stable flameholder for scramjet engine.