  This paper presents eigensolution and non-modal analyses for immersed
boundary methods (IBMs) based on volume penalization for the linear advection
equation. This approach is used to analyze the behavior of flux reconstruction
(FR) discretization, including the influence of polynomial order and
penalization parameter on numerical errors and stability. Through a
semi-discrete analysis, we find that the inclusion of IBM adds additional
dissipation without changing significantly the dispersion of the original
numerical discretization. This agrees with the physical intuition that in this
type of approach, the solid wall is modelled as a porous medium with vanishing
viscosity. From a stability point view, the selection of penalty parameter can
be analyzed based on a fully-discrete analysis, which leads to practical
guidelines on the selection of penalization parameter. Numerical experiments
indicate that the penalization term needs to be increased to damp oscillations
inside the solid (i.e. porous region), which leads to undesirable time step
restrictions. As an alternative, we propose to include a second-order term in
the solid for the no-slip wall boundary condition. Results show that by adding
a second-order term we improve the overall accuracy with relaxed time step
restriction. This indicates that the optimal value of the penalization
parameter and the second-order damping can be carefully chosen to obtain a more
accurate scheme. Finally, the approximated relationship between these two
parameters is obtained and used as a guideline to select the optimum penalty
terms in a Navier-Stokes solver, to simulate flow past a cylinder.
