Energy-decaying spectral scheme with adaptive time steps for diffusive–viscous wave equation on three-dimensional unbounded domain

This work is devoted to designing an efficient numerical scheme for the diffusive–viscous wave equation defined on three-dimensional unbounded domain. For this purpose, the spatial and temporal components are approximated by Hermite–Galerkin spectral method and adaptive time-stepping Crank–Nicolson method, respectively. The main advantages of the proposed scheme are that (i) the time step sizes are adaptively controlled based on the energy variation, which can highly enhance the computational efficiency; (ii) our method is energy-decaying at the fully-discrete level, which meets the L²-energy dissipative nature of the diffusive–viscous wave equation; (iii) the Hermite functions with scaling factor naturally match the unbounded domain of the equation, which avoids the creation of artificial reflections and truncation errors. Finally, a series of numerical experiments is conducted to demonstrate the effectiveness of our scheme.