ENO-Based High-Order Data-Bounded and Constrained Positivity-Preserving Interpolation

TL;DR

This paper introduces a novel high-order ENO-based interpolation method that guarantees data boundedness and positivity preservation, crucial for scientific applications like weather prediction and combustion simulations, especially with non-equispaced data points.

Contribution

The paper develops a new high-order interpolation algorithm that ensures property preservation, including data boundedness and constrained positivity, with theoretical guarantees and practical demonstrations.

Findings

The algorithm guarantees property preservation in 1D and 2D examples.

It provides theoretical conditions for data boundedness and positivity.

The method performs well on non-equispaced data points.

Abstract

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.