Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by Chebyshev collocation method

摘要

We present a simple technique for avoiding physically spurious eigenmodesthat often occur in the solution of hydrodynamic stability problems by theChebyshev collocation method. The method is demonstrated on the solution of theOrr-Sommerfeld equation for plane Poiseuille flow. Following the standardapproach, the original fourth order differential equation is factorised intotwo second-order equations using a vorticity-type auxiliary variable withunknown boundary values which are then eliminated by a capacitance matrixapproach. However the elimination is constrained by the conservation of thestructure of matrix eigenvalue problem, it can be done in two basicallydifferent ways. A straightforward application of the method results in a coupleof physically spurious eigenvalues which are either huge or close to zerodepending on the way the vorticity boundary conditions are eliminated. The zeroeigenvalues can be shifted to any prescribed value and thus removed by a slightmodification of the second approach.