On a stabilization procedure for the parabolic stability equations

The numerical-stability consequences of the remaining ellipticity in the Parabolic Stability Equations (PSE) are studied. The analysis of Li and Malik of the constant-coefficient Navier-Stokes equations is extended by a detailed analysis of the parabolizing steps. Dropping of the highest streamwise derivative removes the slowest decaying upstream propagating mode, whereas the fastest remains. This mode can be numerically damped, by use of an implicit discretization of the streamwise derivative and a large enough streamwise step size. Suggestions of how to make the equations well-posed by the addition of a term proportional to the truncation error of the implicit scheme are given. This term is easy to implement, does not change the order of approximation and removes the step-size restriction. An explicit formula for the critical step size is also derived, in the modified equations, which shows that the equations are completely stabilized for a properly chosen stabilization parameter.