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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=660710
• 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).**

拟议计划的目的是为应用科学和工程中通常解决的问题提供新的数值技术。我的主要重点是为涉及边界和接口的问题开发笛卡尔网格方法。也就是说，问题的几何形状是在沉浸式设置中定义的，例如，一个级别的设置函数。和任意集合，-2-施加接口跳跃条件，-3-为施加边界条件而开发主动惩罚方法。此外，我正在开发与保护法和保存结构的离散化联系的计划的一面。在这里，我建议通过（分析）将方程式转换为保护法来离散ODE或PDE。然后，可以通过例如有限体积方法来通过精确的离散保护来解决该保护定律。这些新方案具有高度理想的长期非线性稳定性。一方面，我计划应用正在开发的技术来解决工程中的重要问题，例如单个（Euler eq。）和多相流（Navier-Stokes），弹性，流体结构相互作用和电磁作用的流体力学。另一方面，我正在开发基本技术，例如任意集，指数分辨率方案，结构保存数值方法的演变。 ***我计划强调3D计算，以便能够解决现实的应用程序，并开发笛卡尔网格方法，原则上可以将其足够通用，以适应任何类型的网格。该提案的总体目标是设计新颖，一般且易于实施的数值方法，重点是准确性（高阶），稳定性（例如结构提供的计划）和效率（线性的指数分辨率方法时间）。**