  In this work, we design and investigate contrast-independent partially
explicit time discretizations for wave equations in heterogeneous high-contrast
media. We consider multiscale problems, where the spatial heterogeneities are
at subgrid level and are not resolved. In our previous work, we have introduced
contrast-independent partially explicit time discretizations and applied to
parabolic equations. The main idea of contrast-independent partially explicit
time discretization is to split the spatial space into two components: contrast
dependent (fast) and contrast independent (slow) spaces defined via multiscale
space decomposition. Using this decomposition, our goal is further
appropriately to introduce time splitting such that the resulting scheme is
stable and can guarantee contrast-independent discretization under some
suitable (reasonable) conditions. In this paper, we propose
contrast-independent partially explicitly scheme for wave equations. The
splitting requires a careful design. We prove that the proposed splitting is
unconditionally stable under some suitable conditions formulated for the second
space (slow). This condition requires some type of non-contrast dependent space
and is easier to satisfy in the "slow" space. We present numerical results and
show that the proposed methods provide results similar to implicit methods with
the time step that is independent of the contrast.
