Uniform Polynomial Stability of a Midpoint-Staggered Finite-Difference Scheme for a Wave Equation With Dynamic Boundary Control

Abstract

A structure-preserving midpoint-staggered finite-difference scheme is analyzed for a one-dimensional wave equation coupled to a first-order dynamic boundary controller. The continuous system has the polynomial energy-decay rate <inline-formula> <tex-math notation="LaTeX">${t}^{\text {-1}}$ </tex-math></inline-formula>, and the main question is whether this rate is preserved uniformly under spatial semi-discretization despite high-frequency numerical effects. The scheme combines nodal variables with midpoint averages and edge differences and preserves the exact discrete dissipation identity. Its staggered structure also yields an exact frequency-domain edge-energy propagation law, avoiding a discretization-dependent high-frequency remainder. We establish a quadratic high-frequency resolvent bound uniformly with respect to the discretization parameter and consequently obtain uniform <inline-formula> <tex-math notation="LaTeX">${t}^{\text {-1}}$ </tex-math></inline-formula> energy decay for the semi-discrete family. Numerical spectra are consistent with the corresponding inverse-square high-frequency behavior as the discretization is refined. No spectral filtering or artificial viscosity is required.