Application of the Crank–Nicolson time integrator to viscoelastic wave equations with boundary feedback damping

Abstract

Abstract We study the numerical solutions for a boundary feedback mechanism on torsional vibrations of a homogeneous viscoelastic rod. The problem is discretized by the Crank–Nicolson scheme based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a two-step way, a convolution quadrature formula, constructed again from the trapezoidal rule, is used to approximate the integral term. Error estimates of the numerical scheme are derived in the $ l^{\infty }_{t}(0,~\infty ;~L^{2}(0,~1)) $ norm. Results from some numerical experiment are presented.