We study the linear stability of high speed flows on compression ramps where experimental and computational evidence suggests presence of flow characteristics that deviate from the two-dimensional laminar behavior. Due to the presence of separated boundary layers, a global stability analysis framework is necessitated. To understand the stability of high speed flows on realistic geometries, we develop an unstructured finite-volume based discretization for the compressible Navier-Stokes equations linearized in conserved variables. We work with conserved variables because their fluxes are continuous across shocks and the linearized equations lend themselves to a discretization employed to compute the base flow-field. In this paper, we focus on obtaining the eigensolutions that describe the late-time perturbation dynamics of the linear system. After verifying the solver with several flow-cases reported in literature, we solve the eigenvalue problem that describes the modal growth of spanwise harmonic linear perturbations in a laminar supersonic flow on a compression ramp. We compare spanwise striations observed in experimental wall tem-perature measurements to the asymptotically unstable perturbations obtained from the biglobal analysis.