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

针对涉及移动界面、跳跃和守恒量问题的高阶数值方法

• 批准号: RGPIN-2016-04628
• 负责人: Nave, JeanChristophe
• 金额: $ 3.35万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=689379
• 关键词: Toward; Order; Numerical; Methods; Problems

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).**

该计划的目的是为应用科学和工程中常见的问题提供新的数值技术。我的主要重点是发展笛卡尔网格方法的问题，涉及边界和接口。也就是说，问题的几何形状是在浸入式设置中定义的，例如，水平集函数。目前的建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-开发施加边界条件的主动罚函数方法。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化联系起来。在这里，我建议离散常微分方程或偏微分方程的转换（解析）到一个守恒律的方程。这个守恒律可以通过例如有限体积法用精确的离散守恒来求解。这些新方案具有非常理想的长期非线性稳定性。这个5年计划的最终目标是覆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。和多相流（Navier-Stokes），弹性，流体-结构相互作用和电磁学。另一方面，我正在开发基本技术，例如任意集合的演化，指数分辨率方案，结构保持数值方法。* 我计划强调3D计算，以便能够处理现实应用，并开发笛卡尔网格方法，原则上可以适用于任何类型的网格。该提案的首要目标是设计新颖，通用和易于实现的数值方法，重点关注精度（高阶），稳定性（例如结构保持方案）和效率（线性时间中的指数分辨率方法）。