Algorithm xxxx: HiPPIS A High-Order Positivity-Preserving Mapping Software for Structured Meshes

TL;DR

HiPPIS tackles the challenge of mapping data between structured meshes while preserving positivity and data-boundedness. It introduces high-order DBI and PPI interpolation based on adaptive ENO-like stencil selection and Newton polynomials, with explicit bounds controlled by \epsilon_0 and \epsilon_1. The software provides 1D, 2D, and 3D implementations (tensor-product extensions) and demonstrates improved accuracy over spline-based and rescaling methods on smooth problems, with careful handling of extrema and discontinuities. The mapping-error analysis and diverse numerical experiments (including NEPTUNE-inspired, TWP-ICE, and BOMEX scenarios) illustrate practical benefits and limitations, emphasizing the need for adequate resolution to fully exploit high-order, positivity-preserving interpolation in coupled PDE simulations.

Abstract

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.

Figures (7)

• Figure 1: Diagram showing the components of the main module used to build the HiPPIS software.
• Figure 2: Approximation of $f_2(x)$ from $N=17$ uniformly spaced points with different interpolation methods. The top row (Figure \ref{['subfig:heaviside1']}) shows approximation results using PCHIP, MQSI, and DBI. The bottom row (Figure \ref{['subfig:heaviside2']}) shows approximation results using PPI with $d=4, 8$ and $\epsilon_{0}=1.0, 0.01$. An enlarged version of the region in the red rectangle is shown on the right of each row. The value of $\epsilon_{1}$ is set to $1.0$.
• Figure 3: Approximation of $f_3(x)$ from $N=17$ uniformly spaced points with different interpolation methods. The top row (Figure \ref{['subfig:GelbT1']}) shows approximation results using PCHIP, MQSI, and DBI. The bottom row (Figure \ref{['subfig:GelbT2']}) shows approximation results using PPI with $d=4, 8$ and $\epsilon_{0}=1.0, 0.01$. An enlarged version of the region in the red rectangle is shown on the right of each row. The value of $\epsilon_{1}$ is set to $1.0$.
• Figure 4: Approximation of $f_5(x,y)$ from $N \times N =17^2$ uniformly spaced points with different interpolation methods. The parameter $st$ is set to $2$. The red ellipses highlight examples regions with large oscillations. The surfaces in the top row Figures \ref{['subfig:heaviside2DPCHIP']} and \ref{['subfig:heaviside2DMQSI']} are obtained using PCHIP and MQSI. Figures \ref{['subfig:heaviside2DP4_1']} ($\mathcal{P}_{4}$, $\epsilon_{0}=\epsilon_{1}=1.0$), \ref{['subfig:heaviside2DP4_2']} ($\mathcal{P}_{4}$, $\epsilon_{0}=\epsilon_{1}=10^{-4}$), \ref{['subfig:heaviside2DP8_1']} ($\mathcal{P}_{8}$, $\epsilon_{0}=\epsilon_{1}=1.0$), and \ref{['subfig:heaviside2DP8_2']} ($\mathcal{P}_{8}$, $\epsilon_{0}=\epsilon_{1}=10^{-4}$) are obtained using PPI.
• Figure 5: Approximation of $f_5(x,y)$ from $N \times N =17^2$ uniformly spaced points with different interpolation methods. The parameter $st$ is set to $2$. The red ellipses highlight examples regions with large oscillations. The surfaces in the top row Figures \ref{['subfig:surface2PCHIP']} and \ref{['subfig:surface2MQSI']} are obtained using PCHIP and MQSI. Figures \ref{['subfig:surface2P4_1']} ($\mathcal{P}_{4}$, $\epsilon_{0}=\epsilon_{1}=1.0$), \ref{['subfig:surface2P4_2']} ($\mathcal{P}_{4}$, $\epsilon_{0}=\epsilon_{1}=10^{-4}$), \ref{['subfig:surface2P8_1']} ($\mathcal{P}_{8}$, $\epsilon_{0}=\epsilon_{1}=1.0$), and \ref{['subfig:surface2P8_2']} ($\mathcal{P}_{8}$, $\epsilon_{0}=\epsilon_{1}=10^{-4}$) are obtained using PPI.