On shock-capturing schemes for the linear stability analysis of high-speed flows

A novel implementation of weighted essentially non-oscillatory (WENO) schemes is presented to solve eigenvalue problems resulting from numerical discretization of the compressible linearized Navier–Stokes equations within a matrix-forming framework. In boundary layer profiles featuring a discontinuity, agreement of the eigenvalues delivered by the proposed method and those found in the literature, calculated using the linearized Rankine–Hugoniot boundary condition, is excellent at the physically most significant conditions of peak amplification. Moreover, the present WENO approach delivers accurate eigenfunctions, drastically reducing oscillations that appear in the amplitude functions of linear perturbations when spatial discretization is performed using standard finite-difference methods. In the absence of discontinuities, accurate recovery of convective instabilities is demonstrated in open and closed shear flows, at the nominal theoretical convergence rate of the WENO method.