Unconditionally energy-stable mapped Gegenbauer spectral Galerkin method for diffusive–viscous wave equation in unbounded domains

The diffusive–viscous wave equation plays a significant role in investigating the attenuation of seismic wave propagating in fluid-saturated medium. This paper focuses on the design of unconditionally energy-stable spectral method for the diffusive–viscous wave equation defined in unbounded domains R^d(d=2,3). We develop such a spectral method by using mapped Gegenbauer functions for the spatial approximation and Crank–Nicolson scheme for the temporal discretization. Then we show that the fully-discrete method satisfies the discrete energy-dissipation law without any restriction on the time step size. To achieve an efficient implementation, the matrix diagonalization procedure is employed to solve the linear systems for d=2,3. Finally, we carry out numerical examples in 3D case to demonstrate the good behavior of our method.