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@article{Michael:2008,
author = {Michael, S Broadhurst and Spencer, J Sherwin},
journal = {Applied Numerical Mathematics},
pages = {1017--1029},
title = {The Parabolised Stability Equations for 3D-Flows: Implementation and Numerical Stability},
url = {http://www2.imperial.ac.uk/ssherw/spectralhp/papers/APNUM-BrSh-07.pdf},
volume = {58},
year = {2008}
}

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TY  - JOUR
AB  - The numerical implementation of the parabolised stability equations (PSE) using a spectral/hp-element discretisation is considered, and the numerical stability of the governing equations is presented. Analogous to the primitive variable form of the two-dimensional PSE, the equations are ill-posed; although choosing an Euler implicit scheme in the streamwise z-direction yields a stable scheme for sufficiently large step sizes (Δz>1/|β|, where β is the streamwise wavenumber). The source of the instability is a residual ellipticity that remains in the equations, and presents itself as an upstream propagating acoustic wave. Neglecting this term relaxes the lower limit on the step-size restriction. The θ-scheme is also considered, allowing the step-size restriction of the scheme to be determined. The explicit scheme is always unstable, whereas neglecting the pressure gradient term shows stable eigenspectra for θ>=0.5
AU  - Michael,S Broadhurst
AU  - Spencer,J Sherwin
EP  - 1029
PY  - 2008///
SP  - 1017
TI  - The Parabolised Stability Equations for 3D-Flows: Implementation and Numerical Stability
T2  - Applied Numerical Mathematics
UR  - http://www2.imperial.ac.uk/ssherw/spectralhp/papers/APNUM-BrSh-07.pdf
UR  - http://dx.doi.org/10.1016/j.apnum.2007.04.016
VL  - 58
ER  -

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