A Numerical Study of Spectral Differentiation Methods: a Comparison of Differentiation Matrix and Trigonometric Transformation Approaches

Spectral methods are found to provide exponential convergence for smooth problems, however their application to these problems is limited by conditioning issues in finite precision arithmetic. An analysis of numerical stability in spectral differentiation is carried out in the work comparing matrix-based method with the trigonometric transformation based approach. The growth of condition number, backward stability and propagation of rounding error together with the stability of coordinate transformation method is analyzed in the work. Rigorous bounds on the error growth in both the approaches are established, numerical experiment carried out validate the theoretical approach bring out the significant of the trigonometric approach in spectral differentiation. The results have significance importance in high-order spectral methods and provide a practical approach for selecting appropriate differentiation strategies for a specific problem.