  A new second-order method for approximating the compressible Euler equations
is introduced. The method preserves all the known invariant domains of the
Euler system: positivity of the density, positivity of the internal energy and
the local minimum principle on the specific entropy. The technique combines a
first-order, invariant domain preserving, Guaranteed Maximum Speed method using
a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy
consistent, high-order method. Invariant domain preserving auxiliary states,
naturally produced by the GMS-GV1 method, are used to define local bounds for
the high-order method which is then made invariant domain preserving via a
convex limiting process. Numerical tests confirm the second-order accuracy of
the new GMS-GV2 method in the maximum norm, where 2 stands for second-order.
The proposed convex limiting is generic and can be applied to other
approximation techniques and other hyperbolic systems.
