  We develop a novel multiscale model of interface motion for the
Rayleigh-Taylor instability (RTI) and Richtmyer-Meshkov instability (RMI) for
two-dimensional, inviscid, compressible flows with vorticity, which yields a
fast-running numerical algorithm that produces both qualitatively and
quantitatively similar results to a resolved gas dynamics code, while running
approximately two orders of magnitude (in time) faster. Our multiscale model is
founded upon a new compressible-incompressible decomposition of the velocity
field $u=v+w$. The incompressible component $w$ of the velocity is also
irrotational and is solved using a new asymptotic model of the Birkhoff-Rott
singular integral formulation of the incompressible Euler equations, which
reduces the problem to one spatial dimension. This asymptotic model, called the
higher-order $z$-model, is derived using small nonlocality as the asymptotic
parameter, allows for interface turn-over and roll-up, and yields a significant
simplification for the equation describing the evolution of the amplitude of
vorticity. This incompressible component $w$ of the velocity controls the small
scale structures of the interface and can be solved efficiently on fine grids.
Meanwhile, the compressible component of the velocity $v$ remains continuous
near contact discontinuities and can be computed on relatively coarse grids,
while receiving subgrid scale information from $w$. We first validate the
incompressible higher-order $z$-model by comparison with classical RTI
experiments as well as full point vortex simulations. We then consider both the
RTI and the RMI problems for our multiscale model of compressible flow with
vorticity, and show excellent agreement with our high-resolution gas dynamics
solutions.
