Computation of frequency responses for linear time-invariant PDEs on a compact interval

TL;DR

This paper introduces a novel spectral collocation method for computing frequency responses of linear time-invariant PDEs on a compact interval, avoiding discretization and improving accuracy and stability.

Contribution

It develops an alternative spectral collocation approach that recasts the problem as a boundary value problem, bypassing finite-dimensional approximations and enhancing numerical stability.

Findings

Achieves machine precision accuracy in frequency response calculations.

Avoids numerical instabilities in high-order differential systems.

Simplifies boundary condition implementation.

Abstract

We develop mathematical framework and computational tools for calculating frequency responses of linear time-invariant PDEs in which an independent spatial variable belongs to a compact interval. In conventional studies this computation is done numerically using spatial discretization of differential operators in the evolution equation. In this paper, we introduce an alternative method that avoids the need for finite-dimensional approximation of the underlying operators in the evolution model. This method recasts the frequency response operator as a two point boundary value problem and uses state-of-the-art automatic spectral collocation techniques for solving integral representations of the resulting boundary value problems with accuracy comparable to machine precision. Our approach has two advantages over currently available schemes: first, it avoids numerical instabilities encountered in systems with differential operators of high order and, second, it alleviates difficulty in implementing boundary conditions. We provide examples from Newtonian and viscoelastic fluid dynamics to illustrate utility of the proposed method.