  Using standard intrusive techniques when solving hyperbolic conservation laws
with uncertainties can lead to oscillatory solutions as well as nonhyperbolic
moment systems. The Intrusive Polynomial Moment (IPM) method ensures
hyperbolicity of the moment system while restricting oscillatory over- and
undershoots of specified bounds. In this contribution, we derive a second-order
discretization of the IPM moment system which fulfills the maximum principle.
This task is carried out by investigating violations of the specified bounds
due to the errors from the numerical optimization required by the scheme. This
analysis gives weaker conditions on the entropy that is used, allowing the
choice of an entropy which enables choosing the exact minimal and maximal value
of the initial condition as bounds. Solutions calculated with the derived
scheme are nonoscillatory while fulfilling the maximum principle. The
second-order accuracy of our scheme leads to significantly reduced numerical
costs.
