High-Order Invariant Domain Preserving Approximations of Multiphysics Systems of Conservation Equations

本项目的目标是在一维、二维和三维空间的非结构网格上构造非线性守恒方程的稳健逼近技术。这类问题涉及工程中的许多领域(机械、航空航天、核能、海洋等)。在科学方面(地球物理、天体物理学)。一套新的用于在现实环境中求解复杂的非线性守恒方程的新的稳健逼近技术也将有利于涉及混合双曲性和其他物理效应(如扩散和弥散)的模型的大量应用。该项目的成果将通过研究生课程、学生辅导、研讨会、会议报告、出版物以及与在美国各机构工作的同事的直接合作来传播。在这个项目中开发的材料将被纳入德克萨斯A&A&M数学系每两年上一次的关于守恒方程的高级课程。在这个项目中，稳健性意味着所提出的方法保证提供满足物理和热力学约束的解(通常基于准凹泛函或凹泛函)。稳健性还意味着，即使网格不够细，不能在某个渐近范围内(即，必须避免锁定)，解的某些渐近性质也可以被近似保持。在文献中，这些类型的方法通常被认为是渐近保持的。最后，我们考虑的算法必须非常少地依赖于手头的空间离散化，并且足够简单，以供几乎不了解数值分析和非线性系统的数学结构的用户编程。该项目将在很大程度上依赖于PI最近建立的坚实的理论基础，并将围绕三个目标进行组织：(I)构建关于多项式次数、网格结构和极少(如果有的话)可调参数的非线性双曲型系统的算法；(Ii)构建用于混合双曲性和其他物理效应(如扩散和弥散)的模型的稳健的近似技术；(Iii)构建技术，确保即使在物理知识不完全的情况下，离散近似也能满足现实的物理边界和热力学不等。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The objective of this project is to construct robust approximation techniques for nonlinear conservation systems on unstructured meshes in one, two, and three space dimensions. This class of problems touches many fields in engineering (mechanical, aerospace, nuclear, ocean, etc.) and in sciences (geophysics, astrophysics). A new set of novel robust approximation techniques for solving complex nonlinear conservation equations in realistic settings will also benefit numerous applications that involve models mixing hyperbolicity with other physical effects such as diffusion and dispersion. The results of this project will be disseminated through graduate classes, mentoring of students, seminars, conference presentations, publications, and direct collaborations with colleagues working at various US institutions. The material developed in this project will be incorporated in an advanced class on conservation equations that is given every two years in the Department of Mathematics at Texas A&M.In this project, robustness means that the proposed methods are guaranteed to deliver solutions that satisfy physical and thermodynamical constraints (in general based on quasi-concave or concave functionals). Robustness also means that some asymptotic properties of the solution may be preserved by the approximation even if the mesh is not fine enough to be in some asymptotic range (i.e., locking must be avoided). These types of methods are often said to be asymptotic preserving in the literature. Finally, the algorithms we have in mind must depend very little on the space discretization at hand and be simple enough to be programed by users with very little know-how in numerical analysis and on the mathematical structure of the nonlinear system. The project will heavily rely on the solid theoretical foundations recently established by the PIs and will be organized around three objectives: (i) construct algorithms for nonlinear hyperbolic systems that are robust with respect to the polynomial degree, the mesh structure, and have very few (if any) tuning parameters; (ii) construct approximation techniques that are robust for models mixing hyperbolicity with other physical effects such as diffusion and dispersion; (iii) construct techniques that will guarantee that realistic physical bounds and thermodynamical inequalities are satisfied for the discrete approximation even when incomplete knowledge of the physics is available.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.