An efficient Hermite spectral solver for wave equations in micro-heterogeneous porous media

We propose an efficient Hermite spectral method for solving the initial-value problem of an integro-differential partial differential equation that models wave propagation in micro-heterogeneous porous media. Leveraging the properties of tempered fractional calculus, the original integro-differential equation with a specific memory kernel is first reformulated as an equivalent time-fractional PDE. Using the Faedo-Galerkin technique, we establish the well-posedness of the reformulated model in an unbounded domain. For numerical discretization, the spatial derivative is treated with the Hermite spectral method, and the resulting semi-discrete scheme is shown to be unconditionally stable and spectrally accurate. In time, the second-order term is discretized by a central difference scheme, while the tempered Caputo fractional derivative is approximated via the weighted and shifted Grünwald-Letnikov formula. Numerical experiments not only validate the theoretical stability and convergence analysis but also confirm the second-order temporal accuracy of the full discretization. Furthermore, the simulations reveal a rich variety of dynamical behaviors exhibited by the model.