Stabilization of the Wave Equation by Direct Fourier Filtering
A boundary-controlled wave-equation model comparing direct Fourier filtering with an order-reduced discretization.
Overview
This Demonstration models a one-dimensional wave controlled by damping at its free end. The live controls choose an initial configuration and its location, select finite differences, finite elements, or order reduction, and vary the node count, viscous damping, boundary damping, and direct-Fourier-filter cutoff.
Mathematical model
The model uses distributed damping $k_1$ and boundary feedback $k_3$. In standard discrete models, high-frequency eigenvalues drift toward the imaginary axis as $h\to0$; direct Fourier filtering removes this poorly damped part of the spectrum, while order reduction retains the relevant stability behavior without filtering:
\[\begin{aligned} y_{tt} + k_1 y_t - y_{xx} &= 0, \\ y(0,t) &= 0, \\ y_x(L,t) + k_3 y_t(L,t) &= 0. \end{aligned}\]Research project
This Demonstration is part of Observability-Preserving Discretizations, which studies numerical models that retain reliable boundary sensing and feedback control. It is the collection’s most elementary example of how a spectral filter and a structure-preserving discretization address the same control-theoretic problem.
Reference
Walterman, Jacob, Ahmet Kaan Aydin, Matthew Poynter, and Ahmet Özkan Özer. 2024. Stabilization of the Wave Equation by Direct Fourier Filtering. Wolfram Demonstrations Project.