The pseudospectral method for solving differential eigenvalue problems
Journal of Computational Physics
Generation of Pseudospectral Differentiation Matrices I
SIAM Journal on Numerical Analysis
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
A MATLAB differentiation matrix suite
ACM Transactions on Mathematical Software (TOMS)
Computation of Gauss-type quadrature formulas
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. V: quadrature and orthogonal polynomials
Essentials of Numerical Analysis with Pocket Calculator Demonstrations
Essentials of Numerical Analysis with Pocket Calculator Demonstrations
Journal of Computational and Applied Mathematics
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We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green functions, building on the successful use of split-Gauss-type quadrature schemes. Here we adapt the method for eigenvalue problems, in particular the Orr-Sommerfeld equation of hydrodynamic stability theory. Use of the Green function for the viscous part of the problem reduces the fourth-order ordinary differential equation to an integro-differential equation which we then discretize using the split-Gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators. As the latter are only second-order the resulting discrete equations are much more stable than those obtained from the original differential equation. This permits us to obtain results for the standard test problem (plane Poiseuille flow at unit streamwise wavenumber and Reynolds number 10000) that we believe are the most accurate to date.