A spectral Chebyshev method for linear stability analysis of one-dimensional exact solutions of gas dynamics

We present a spectral numerical method for solving one-dimensional systems of partial differential equations (PDEs) which arise from linearization of the Euler equations about an exact solution depending on space and time. A two-domain Chebyshev collocation method is used. Matching of quantities is performed in the space of characteristic variables as suggested by Kopriva [Appl. Numer. Math. 2 (1986) 221; J. Comput. Phys. 125 (1996) 244]. Time-dependent boundary conditions are handled following an approach proposed by Thompson [J. Comput. Phys. 68 (1987) 1; 89 (1990) 439]. An exact numerical stability analysis valid for any explicit three-step third-order non-degenerate Runge–Kutta scheme is provided. The numerical method is tested against exact solutions for the three fundamental modes of a compressible flow (entropy, vorticity and acoustic modes).