  An invariant-region-preserving (IRP) limiter for multi-dimensional hyperbolic
conservation law systems is introduced, as long as the system admits a global
invariant region which is a convex set in the phase space. It is shown that the
order of approximation accuracy is not destroyed by the IRP limiter, provided
the cell average is away from the boundary of the convex set. Moreover, this
limiter is explicit, and easy for computer implementation. A generic algorithm
incorporating the IRP limiter is presented for high order finite volume type
schemes. For arbitrarily high order discontinuous Galerkin (DG) schemes to
hyperbolic conservation law systems, sufficient conditions are obtained for
cell averages to remain in the invariant region provided the projected
one-dimensional system shares the same invariant region as the full
multi-dimensional hyperbolic system {does}. The general results are then
applied to both one and two dimensional compressible Euler equations so to
obtain high order IRP DG schemes. Numerical experiments are provided to
validate the proven properties of the IRP limiter and the performance of IRP DG
schemes for compressible Euler equations.
