Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay

This paper presents a fully implicit finite difference scheme for the numerical approximation of a wave equation featuring strong damping and a distributed delay term. The discretization employs second-order accurate approximations in both time and space. Although implicit, the scheme does not ensure unconditional stability due to the nonlocal nature of the delayed damping. To address this, we perform a stability analysis based on Rouché’s theorem from complex analysis and derive a sufficient condition for asymptotic stability of the discrete system. The resulting criterion highlights the interplay among the discretization parameters, the damping coefficient, and the delay kernel. Two quadrature techniques, the composite trapezoidal rule (CTR) and the Gaussian quadrature rule (GQR), are employed to approximate the convolution integral. Numerical experiments validate the theoretical predictions and illustrate both stable and unstable dynamics across different parameter regimes.