  In this paper, we develop a computational multiscale to solve the parabolic
wave approximation with heterogeneous and variable media. Parabolic wave
approximation is a technique to approximate the full wave equation. One benefit
of the method is that: one wave propagation direction can be taken as an
evolution direction, and we then can discretize it using a classical scheme
like Backward Euler. Consequently, we obtain a set of quasi-gas-dynamic (QGD)
models with different heterogeneous permeability fields. Then, we employ
constraint energy minimization generalized multiscale finite element method
(CEM-GMsFEM) to perform spatial discretization for the problem. The resulting
system can be solved by combining the central difference in time evolution. Due
to the variable media, we apply the technique of proper orthogonal
decomposition (POD) to further the dimension of the problem and solve the
corresponding model problem in the POD space instead of in the whole multiscale
space spanned by all possible multiscale basis functions. We prove the
stability of the full discretization scheme and give the convergence analysis
of the proposed approximation scheme. Numerical results verify the
effectiveness of the proposed method.
