We present a simple technique for avoiding physically spurious eigenmodes
that often occur in the solution of hydrodynamic stability problems by the
Chebyshev collocation method. The method is demonstrated on the solution of the
Orr-Sommerfeld equation for plane Poiseuille flow. Following the standard
approach, the original fourth order differential equation is factorised into
two second-order equations using a vorticity-type auxiliary variable with
unknown boundary values which are then eliminated by a capacitance matrix
approach. However the elimination is constrained by the conservation of the
structure of matrix eigenvalue problem, it can be done in two basically
different ways. A straightforward application of the method results in a couple
of physically spurious eigenvalues which are either huge or close to zero
depending on the way the vorticity boundary conditions are eliminated. The zero
eigenvalues can be shifted to any prescribed value and thus removed by a slight
modification of the second approach.Comment: 10 pages, 1 figure, minor revision, to appear in J. Comp. Phy