Toward High-Order Numerical Methods for Problems Involving Moving Interfaces, Jumps and Conserved Quantities

拟议方案的目的是为应用科学和工程学中常见的问题提供新的数值技术。我的主要关注点是为涉及边界和界面的问题开发笛卡尔网格方法。也就是说，通过例如水平集函数在沉浸式环境中定义问题的几何。*当前的提议集中于开发高阶笛卡尔网格方法，用于：-1-曲线、曲面和任意集合的演化，-2-施加界面跳跃条件，以及-3-开发用于施加边界条件的主动惩罚方法。此外，我正在开发我的程序的一个方面，将守恒定律和结构保持离散联系起来。在这里，我建议通过将方程(解析地)转化为守恒定律来离散常微分方程组或偏微分方程组。然后，这个守恒律可以用精确的离散守恒定律来求解，例如有限体积法。这些新方案具有非常理想的长期非线性稳定性特性。*这个5年计划的最终目标是涵盖从应用到理论的广泛领域。一方面，我计划应用我正在开发的技术来解决工程中的重要问题，例如单(欧拉方程)的流体力学。以及多相流(纳维斯托克斯)、弹性、流固耦合和电磁。另一方面，我正在发展基本的技术，如任意集合的演化，指数分辨格式，结构保持的数值方法。*我计划强调3D计算，以便能够处理实际应用，并开发原则上足够通用的笛卡尔网格方法，以适应任何类型的网格。这一提议的首要目标是设计新颖、通用、易于实施的数值方法，重点关注精度(高阶)、稳定性(例如结构保持格式)和效率(线性时间内的指数分辨方法)。

The aim of the proposed program is to provide new numerical techniques for problems commonly tackled in the applied sciences and engineering. My main focus is to develop Cartesian grid methods for problems involving boundaries and interfaces. That is, the geometry of the problem is defined in an immersed setting by for example, a level set function.***The current proposal focuses on developing high-order Cartesian grid methods for: -1- the evolution of curves, surfaces, and arbitrary sets, -2- imposing interface jump conditions, and -3- developing active penalty methods for imposing boundary conditions. In addition, I am developing a side of my program linking conservation laws and structure-preserving discretizations. Here, I propose to discretize ODEs or PDEs by transforming (analytically) the equation into a conservation law. This conservation law can then be solved with exact discrete conservation by for example, the finite-volume method. These new schemes possess highly desirable long-term non-linear stability properties.***The ultimate goal of this 5-year program is to cover a wide swath from applications to theory. On the one hand, I plan to apply the techniques I am developing to tackle important problems from engineering e.g. fluid mechanics of single- (Euler eq.) and multi-phase flows (Navier-Stokes), elasticity, fluid-structure interaction, and electromagnetism. On the other hand, I am developing fundamental techniques e.g. evolution of arbitrary sets, exponential-resolution schemes, structure-preserving numerical methods. ***I plan on emphasizing 3D computations so as to be able to tackle realistic applications, and develop Cartesian grid methods that could in principle be general enough to be adapted to any type of mesh. The overarching objective of this proposal is to design numerical methods that are novel, general, and easy to implement with a focus on accuracy (high-order), stability (e.g. structure-preserving schemes), and efficiency (exponential-resolution methods in linear time).**