Adaptive Partition of Unity Interpolation Method with Moving Patches

TL;DR

This paper explores an adaptive partition of unity interpolation method with moving patches for accurately interpolating functions with sharp gradients, especially in two-medium problems and evolving vector fields, with applications in volume tracking and multiphase flow.

Contribution

It introduces a dynamic patch-shifting technique for the partition of unity method to better handle evolving functions with sharp features.

Findings

High accuracy volume conservation in 1D and 2D tests

Effective tracking of evolving local profiles

Applicable to multiphase flow modeling

Abstract

The adaptive partition of unity interpolation method, introduced by Aiton and Driscoll, using Chebyshev local interpolants, is explored for interpolating functions with sharp gradients representing two-medium problems. For functions that evolve under vector fields, the partition of unity patches (covers) can be shifted and resized to follow the changing dynamics of local profiles. The method is tested for selected 1D and 2D two-medium problems with linear divergence-free vector fields. In those cases, the volume fraction in each patch contributing to volume conservation throughout the domain can be kept in high accuracy down to machine precisions. Applications that could benefit from the method include volume tracking and multiphase flow modeling.