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

本项目侧重于半拉格朗日方法的高阶推广，称为JET格式(在场量传输方面)和梯度增广水平集方法(GALSM，在界面跟踪方面)。这些数值方法通过跟踪沿特征的解的某些导数来实现高精度。从用于在网格点更新解的数据仅位于单个网格单元中的意义上来说，它们是最优的局部的，与方案的顺序无关。此外，基于单元的Hermite插值法的使用产生了一定水平的亚网格分辨率，这使得GALSM能够捕获比网格分辨率更小的结构。本项目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加密和拉格朗日粒子的结合。此外，JET格式被应用于动力学方程，GALSM被应用于Hamilton-Jacobi方程。界面(曲线和曲面)的准确检测、跟踪和计算是许多科学技术领域的重要问题，如：计算流体力学中的气液界面、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。该项目中开发的方法允许高精度地计算跟踪界面以及场量的演变。同时，它们在计算上是高效的，并且非常模块化。此外，它们还有利于捕获小结构。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.