Polynomial interpolation is an important component of many computational problems. In several of these computational problems, failure to preserve positivity when using polynomials to approximate or map data values between meshes can lead to negative unphysical quantities. Currently, most polynomial-based methods for enforcing positivity are based on splines and polynomial rescaling. The spline-based approaches build interpolants that are positive over the intervals in which they are defined and may require solving a minimization problem and/or system of equations. The linear polynomial rescaling methods allow for high-degree polynomials but enforce positivity only at limited locations (e.g., quadrature nodes). This work introduces open-source software (HiPPIS) for high-order data-bounded interpolation (DBI) and positivity-preserving interpolation (PPI) that addresses the limitations of both the spline and polynomial rescaling methods. HiPPIS is suitable for approximating and mapping physical quantities such as mass, density, and concentration between meshes while preserving positivity. This work provides Fortran and Matlab implementations of the DBI and PPI methods, presents an analysis of the mapping error in the context of PDEs, and uses several 1D and 2D numerical examples to demonstrate the benefits and limitations of HiPPIS.

多项式插值是许多计算问题的重要组成部分。在其中一些计算问题中，当使用多项式在网格之间近似或映射数据值时未能保持正性可能会导致负非物理量。目前，大多数基于多项式的增强积极性的方法都是基于样条和多项式缩放。基于样条的方法构建在定义的间隔内为正的插值，并且可能需要求解最小化问题和/或方程组。线性多项式缩放方法允许高次多项式，但仅在有限位置（例如，正交节点）强制执行正性。这项工作引入了用于高阶数据有界插值 (DBI) 和正性保留插值 (PPI) 的开源软件 (HiPPIS)，该软件解决了样条和多项式缩放方法的局限性。HiPPIS 适用于近似和映射物理量，例如网格之间的质量、密度和浓度，同时保持正性。这项工作提供了 DBI 和 PPI 方法的 Fortran 和 Matlab 实现，对偏微分方程背景下的映射误差进行了分析，并使用几个一维和二维数值示例来演示 HiPPIS 的优点和局限性。