High order numerical methods for multiphysics couplings

该项目导致了用于复杂流动和运输问题的高阶间断Galerkin方法的发展。研究人员对不同类型的耦合进行了分析和数值研究，例如具有混相置换和多相流的Stokes方程或Navier-Stokes方程的耦合。在表面区和次表面区之间的界面上施加了物理上有意义的条件。数值分析的数学进展包括证明耦合系统高阶解的存在唯一性，以及严格推导关于网格尺寸和多项式次数的先验误差估计。数值解的验证是通过收敛研究、关于网格大小和多项式次数的适应性以及与其他离散化技术得到的解的比较来完成的。开发了一个适用于二维和三维问题的计算框架。这项工作涉及多物理问题的建模、分析和模拟，例如通过河流和湖泊的地下水污染。当人造物质或有时是自然存在的物质溶解在补给地下水的水中时，地下水就会受到污染。通常，由于地下水与湖泊和河流相连，这些地表水的污染意味着对含水层的污染。研究人员研究了地表流动和地下流动之间的相互作用。发展了精确高效的数值方法，并进行了分析。密集的数值模拟和与实验数据的比较使我们能够进一步了解多物理耦合。这项任务的跨学科性质促进了科学知识在工程和科学界的转移。这个项目通过培训几名研究生和本科生来促进发现和理解，同时促进学习。一项重要的教育活动是为进入10-12年级的高中生提供为期一周的暑期数学学校。S活动的目的是鼓励学生在科学和数学方面追求职业生涯。

The project results in the development of high order discontinuous Galerkin methods for complex flow and transport problems. The investigator analyzes and numerically studies different types of couplings, such as the coupling of Stokes or Navier-Stokes equations with miscible displacement and multiphase flow. Physically meaningful conditions are imposed at the interface between surface region and subsurface region. The mathematical advances in numerical analysis consist of proving existence and uniqueness of high order solutions of coupled systems and of rigorously deriving a priori error estimates with respect to the mesh size and the polynomial degree. Verification of the numerical solutions is performed through convergence studies, adaptivity with respect to the mesh size and the polynomial degree and comparisons with solutions obtained from other discretization techniques. A computational framework valid for two and three-dimensional problems is developed.This work deals with the modeling, analysis and simulation of multiphysics problems arising for instance in groundwater contamination through rivers and lakes. Groundwater becomes contaminated when human-made substances or sometimes naturally occurring substances are dissolved in waters recharging the groundwater. Often, as groundwater is connected with lakes and rivers, the pollution of these surface waters implies the pollution of aquifers. The investigator studies the interaction between surface flow and subsurface flow. Accurate and efficient numerical methods are developed and analyzed. Intensive numerical simulations and comparison with experimental data allow gaining further knowledge on multiphysics couplings. The interdisciplinary nature of the task fosters transfer of scientific knowledge across the engineering and science communities.This project advances discovery and understanding while promoting learning by training several graduate students and undergraduate students. An important educational activity is a week-long Summer Math School offered to high-school students entering grades 10-12. The event?s objective is to encourage students to pursue a career in science and mathematics.