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Small perturbation evolution in compressible Poiseuille flow is contrasted against the incompressible case using direct simulations and non-modal linear analysis. The onset of compressibility effects leads to a profound change in the behaviour of pressure and its interaction with the velocity field. Linear analysis shows that the most significant compressibility outcome is the harmonic coupling between pressure and wall-normal velocity perturbations. Oscillations in normal perturbations can lead to periods of negative production causing suppression of perturbation growth. The extent of the influence of compressibility can be characterized in terms of an effective gradient Mach number (
). Analysis shows that
diminishes as the angle of the perturbation increases with respect to the shear plane. Direct numerical simulations show that streamwise perturbations, which would lead to Tollmien–Schlichting instability in the incompressible case, are completely suppressed in the compressible case and experience the highest
. At the other extreme, computations reveal that spanwise perturbations, which experience negligible
, are entirely unaltered from the incompressible case. Perturbation behaviour at intermediate obliqueness angles is established. Moreover, the underlying pressure–velocity interactions are explicated.
A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate is established under the L2-norm when polynomials of degree atmost r and k are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.
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