  A moving discontinuous Galerkin finite element method with interface
conservation enforcement (MDG+ICE) is developed for solving the compressible
Euler equations. The MDG+ICE method is based on the space-time DG formulation,
where both flow field and grid geometry are considered as independent variables
and the conservation laws are enforced both on discrete elements and element
interfaces. The element conservation laws are solved in the standard
discontinuous solution space to determine conservative quantities, while the
interface conservation is enforced using a variational formulation in a
continuous space to determine discrete grid geometry. The resulting
over-determined system of nonlinear equations arising from the MDG+ICE
formulation can then be solved in a least-squares sense, leading to an
unconstrained nonlinear least-squares problem that is regularized and solved by
Levenberg-Marquardt method. A number of numerical experiments for compressible
flows are conducted to assess the accuracy and robustness of the MDG+ICE
method. Numerical results obtained indicate that the MDG+ICE method is able to
implicitly detect and track all types of discontinuities via interface
conservation enforcement and satisfy the conservation law on both elements and
interfaces via grid movement and grid management, demonstrating that an
exponential rate of convergence for Sod and Lax-Harden shock tube problems can
be achieved and highly accurate solutions without overheating to both
double-rarefaction wave and Noh problems can be obtained.
