There is a clear need for accurate methods for modeling wave fields in
the time-domain in complex geometries. Applications for the  modeling
of time-domain wave fields are, for example, non-destructive testing,
exploration seismology, electromagnetics, and medical ultrasonics.
Historically, several approaches have been explored in the quest for a
stable method that can efficiently resolve wave without excessive
numerical dissipation or dispersion.

A promising approach for accurately approximating wave fields is the
discontinuous Galerkin (DG) method.  The DG method has several
features that make it an attractive candidate for large-scale wave
simulations. The variational formulation reduces each element of the
computational mesh to a subproblem. With the DG method, the
communication between adjacent elements is handled using the numerical
flux. On the other hand, the variational form allows for easier
parallelization of the solver code and the material parameters, the
order of the basis functions, and the length of the time step can be
chosen individually for each subproblem. Of course, there are also
drawbacks when using DG methods. For example, high-order basis
functions force the use of small time steps, which is time consuming.

In this presentation, one of the main goals is to investigate a
non-uniform basis order based on previous theoretical results. The
focus is to determine the relationship between the element size, wave
number, accuracy of the solution, and order of the basis functions. In
the numerical experiments, the wave propagation in acoustic and
elastic media is studied.