  The Equation-Free approach to efficient multiscale numerical computation
marries trusted micro-scale simulations to a framework for numerical
macro-scale reduction -- the patch dynamics scheme. A recent novel patch scheme
empowered the Equation-Free approach to simulate systems containing shocks on
the macro-scale. However, the scheme did not predict the formation of shocks
accurately, and it could not simulate moving shocks. This article resolves both
issues, as a first step in one spatial dimension, by embedding the
Equation-Free, shock-resolving patch scheme within a classic framework for
adaptive moving meshes. Our canonical micro-scale problems exhibit
heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate
many remarkable benefits from the moving patch scheme, including efficient and
accurate macro-scale prediction despite the unknown macro-scale closure.
Equation-free methods are here extended to simulate moving, forming and merging
shocks without a priori knowledge of the existence or closure of the shocks.
Whereas adaptive moving mesh equations are typically stiff, typically requiring
small time-steps on the macro-scale, the moving macro-scale mesh of patches is
typically not stiff given the context of the micro-scale time-steps required
for the sub-patch dynamics.
