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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615195
• 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。我的主要重点是开发笛卡尔网格方法来解决涉及边界和界面的问题。也就是说，问题的几何形状是通过例如水平集函数在浸入式设置中定义的。 目前的提案侧重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化联系起来。在这里，我建议通过将方程（分析地）转换为守恒定律来离散 ODE 或 PDE。然后可以通过例如有限体积方法用精确的离散守恒来求解该守恒定律。这些新方案具有非常理想的长期非线性稳定性能。 这个为期 5 年的项目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，例如单相流（欧拉方程）和多相流（纳维-斯托克斯）的流体力学、弹性、流固耦合和电磁学。另一方面，我正在开发基本技术，例如任意集的演化、指数分辨率方案、结构保持数值方法。 我计划强调 3D 计算，以便能够解决实际应用，并开发原则上足够通用的笛卡尔网格方法，足以适应任何类型的网格。该提案的总体目标是设计新颖、通用且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（线性中的指数分辨率方法）时间）。