  A number of key scientific computing applications that are based upon
tensor-product grid constructions, such as numerical weather prediction (NWP)
and combustion simulations, require property-preserving interpolation.
Essentially non-oscillatory (ENO) interpolation is a classic example of such
interpolation schemes. In the aforementioned application areas, property
preservation often manifests itself as a requirement for either data
boundedness or positivity preservation. For example, in NWP, one may have to
interpolate between the grid on which the dynamics is calculated to a grid on
which the physics is calculated (and back). Interpolating density or other key
physical quantities without accounting for property preservation may lead to
negative values that are nonphysical and result in inaccurate representations
and/or interpretations of the physical data. Property-preserving interpolation
is straightforward when used in the context of low-order numerical simulation
methods. High-order property-preserving interpolation is, however, nontrivial,
especially in the case where the interpolation points are not equispaced. In
this paper, we demonstrate that it is possible to construct high-order
interpolation methods that ensure either data boundedness or constrained
positivity preservation. A novel feature of the algorithm is that the
positivity-preserving interpolant is constrained; that is, the amount by which
it exceeds the data values may be strictly controlled. The algorithm we have
developed comes with theoretical estimates that provide sufficient conditions
for data boundedness and constrained positivity preservation. We demonstrate
the application of our algorithm on a collection of 1D and 2D numerical
examples, and show that in all cases property preservation is respected.
