  Numerical schemes for wave-like systems with small dissipation are often
inaccurate and unstable due to truncation errors and numerical roundoff errors.
Hence, numerical simulations of wave-like systems lacking proper handling of
these numerical issues often fail to represent the physical characteristics of
wave phenomena. This challenge gets even more intricate for multiscale
modelling, especially in multiple dimensions. When using the usual collocated
grid, about two-thirds of the resolved wave modes are incorrect with
significant dispersion. But, numerical schemes on staggered grids (with
alternating variable arrangement) are significantly less dispersive and
preserve much of the wave characteristics. Also, the group velocity of the
energy propagation in the numerical waves on a staggered grid is in the correct
direction, in contrast to the collocated grid. For high accuracy and to
preserve much of the wave characteristics, this article extends the concept of
staggered grids in full-domain modelling to multidimensional multiscale
modelling. Specifically, this article develops 120 multiscale staggered grids
and demonstrates their stability, accuracy, and wave-preserving characteristic
for equation-free multiscale modelling of weakly damped linear waves. But most
characteristics of the developed multiscale staggered grids must also hold in
general for multiscale modelling of many complex spatio-temporal physical
phenomena such as the general computational fluid dynamics.
