Efficient and dissipation-preserving Hermite spectral Galerkin methods for diffusive-viscous wave equations in unbounded domains

The diffusive-viscous wave equation (DVWE) resulting from the diffusive-viscous wave theory is usually used to describe frequency-dependent seismic reflections in fluid-saturated media, leading to a wider application in seismic exploration. In a previous work (Ling and Mao, 2023), we considered the DVWE in unbounded domain to avoid artificial reflections and truncation errors, and then established its well-posedness and regularity as well as developed a Hermite spectral Galerkin approximation for the space discretization. However, for the time discretization, an explicit Runge–Kutta scheme is employed resulting an extremely restricted time step due to the stability. In the present work, to resolve this issue, we develop two second-order implicit (Crank–Nicolson and BDF2) Hermite spectral methods and show that the proposed schemes are unconditionally energy stable and further establish the convergence. Furthermore, for the two-dimensional case, we develop an efficient algorithm with $O\left(N^{d+1}\right)$ (where $N$ is the degree of freedom of each direction and $d$ is the space dimension) computational cost. In particular, we use the matrix diagonalization technique for the cases of constant coefficients while we employ the preconditioned conjugate gradient method for the cases of variable coefficients. Finally we provide several numerical examples to verify our theoretical results and to demonstrate the effectiveness and efficiency along with other good behavior of the proposed schemes.