Stabilization of spectral methods for the analysis of singular systems using piecewise constant basis functions

Abstract

The analysis of initial value problems associated with singular linear dynamic systems using spectral methods is discussed. These methods, though extensively studied in the literature, can be very sensitive to numerical errors and to the presence of noisy input data. Moreover, using piecewise constant functions, like Walsh or Block-Pulse, spectral methods have been shown to be unstable with respect to incorrect initial conditions. In this paper regularization techniques are used to reduce arbitrarily the sensitivity of the spectral methods, and at the same time to stabilize them. Although the regularized spectral methods maintain the structure of the original algorithm, they can also be rewritten in recursive form, leading to the definition of a class of discrete time filters. Finally, some applicative numerical examples are discussed in order to show the effectiveness of the proposed algorithms.