Numerical approximations of flow derivatives using finite-difference schemes in presence of discontinuities, e.g., shock waves, result in spurious numerical oscillations being introduced to the solution. Such numerical artefacts can corrupt the solution or prevent numerical convergence. Shock capturing is one of the approaches used to prevent such numerical oscillations in DNS and LES. A shock capturing approach selects an upwind scheme among several candidate stencils, as in essentially non-oscillatory (ENO) schemes, or uses a convex combination of all the candidate stencils, as in the weighted ENO (WENO) schemes. In this work, fifth order WENO scheme has been implemented to the eigenvalue problem arising from one-dimensional local stability analysis. The results show that it can correctly recover convective instabilities with strong discontinuities. The WENO5 implementation is also shown to significantly reduce the amplitudes of spurious oscillations introduced to eigenfunctions near strong discontinuities. Ongoing work is being performed to implement alternative WENO and flux splitting formulations and to apply these schemes multidimensional stability analysis problems.