Mathematical Nature of Parabolized Stability Equations

Summary

The primitive-variable formulation of Parabolized Stability Equations is ill-posed due to the ellipticity introduced by $\partial {\hat p}/\partial x$ term and marching solution eventually blows up for sufficiently small step size. It is shown that this difficulty can be overcome if the minimum step size is greater than the inverse of the real part of the streamwise wavenumber, α _r An alternative is to drop the $\partial {\hat p}/\partial x$ term, in which case the residual ellipticity is of no consequence for marching computations with much smaller but practical step sizes.