Collaborative Research: Gradient-augmented level set methods and jet schemes

该项目侧重于半拉格朗日方法的高阶推广，称为射流方案（在场量输运的背景下），以及梯度增强水平集方法（GALSM，在界面跟踪的背景下）。这些数值方法通过沿特征跟踪解的某些导数来达到高阶精度。它们是最优局部的，这意味着用于在网格点更新解决方案的数据仅位于单个网格单元中，与方案的顺序无关。此外，使用基于单元格的Hermite插值产生一定程度的子网格分辨率，这使得GALSM可以捕获比网格分辨率更小的结构。本课题的研究重点是新方法的数值分析和并行性能，以及它们与自适应网格细化和拉格朗日粒子的结合。此外，将射流格式应用于动力学方程，将GALSM应用于Hamilton-Jacobi方程。后者导致引入限制器，并提供了梯度增强重新初始化的路径。界面（曲线和曲面）的精确检测、跟踪和计算是许多科学和技术领域的一个重要问题，例如：计算流体动力学中的气液界面、材料中的相变、天气锋面、生物膜的运动、医学成像中的边缘检测、火焰锋面和超音速流动中的激波锋面。本项目开发的方法允许对界面进行计算跟踪，以及场量的演变，具有很高的精度。同时，它们的计算效率很高，而且非常模块化。此外，它们有利于捕获小结构。该项目涉及国际合作，以及研究生和本科生的培训。

This project focuses on high-order generalizations of semi-Lagrangian approaches, called jet schemes (in the context of the transport of field quantities), and the gradient-augmented level set method (GALSM, in the context of interface tracking). These numerical methods achieve high-order of accuracy by tracking certain derivatives of the solution along characteristics. They are optimally local, in the sense that the data used to update the solution at a grid point is located only in a single grid cell, independent of the scheme's order. Moreover, the use of cell-based Hermite interpolations yields a certain level of subgrid resolution, which allows the GALSM to capture structures smaller than the grid resolution. The research in this project focuses on the numerical analysis and parallel performance of the new approaches, as well as their combination with adaptive mesh refinement and with Lagrangian particles. In addition, jet schemes are applied to kinetic equations, and the GALSM is applied to Hamilton-Jacobi equations. The latter results in the introduction of limiters, and provides a path to gradient-augmented re-initialization.The accurate detection, tracking, and computation of interfaces (curves and surfaces) is an important problem in many areas of science and technology, such as: gas-liquid interfaces in computational fluid dynamics, phase transitions in materials, weather fronts, the motion of biological membranes, edge detection in medical imaging, flame fronts, and shock fronts in supersonic flows. The methods developed in this project allow the computational tracking of interfaces, as well as the evolution of field quantities, with high accuracy. At the same time, they are computationally efficient and very modular. Moreover, they are advantageous for the capture of small structures. This project involves an international collaboration, as well as the training of graduate and undergraduate students.