Imperial College London


Faculty of EngineeringDepartment of Aeronautics

Professor of Computational Fluid Mechanics



+44 (0)20 7594 5052s.sherwin Website




313BCity and Guilds BuildingSouth Kensington Campus






BibTex format

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 = {},
volume = {58},
year = {2008}

RIS format (EndNote, RefMan)

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 -
UR -
VL - 58
ER -