  Many multiscale wave systems exhibit macroscale emergent behaviour, for
example, the fluid dynamics of floods and tsunamis. Resolving a large range of
spatial scales typically requires a prohibitively high computational cost. The
small dissipation in wave systems poses a significant challenge to further
developing multiscale modelling methods in multiple dimensions. This article
develops and evaluates two families of equation-free multiscale methods on
novel 2D staggered patch schemes, and demonstrates the power and utility of
these multiscale schemes for weakly damped linear waves. A detailed study of
sensitivity to numerical roundoff errors establishes the robustness of
developed staggered patch schemes. Comprehensive eigenvalue analysis over a
wide range of parameters establishes the stability, accuracy, and consistency
of the multiscale schemes. Analysis of the computational complexity shows that
the measured compute times of the multiscale schemes may be 10^5 times smaller
than the compute time for the corresponding full-domain computation. This work
provides the essential foundation for efficient large-scale simulation of
challenging nonlinear multiscale waves.
