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

本项目的目标是在一维、二维和三维空间上构建非线性守恒系统的鲁棒逼近技术。这类问题涉及工程（机械、航空航天、核、海洋等）和科学（地球物理、天体物理学）的许多领域。一套新的鲁棒近似技术用于在现实环境中求解复杂的非线性守恒方程，也将有利于许多涉及双曲性与其他物理效应（如扩散和弥散）混合模型的应用。该项目的成果将通过研究生课程、学生指导、研讨会、会议演讲、出版物以及与美国各机构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州农工大学数学系每两年开设一次的守恒方程高级课程。在这个项目中，鲁棒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决方案（通常基于准凹或凹泛函）。鲁棒性还意味着，即使网格不够精细，不能在某个渐近范围内（即必须避免锁定），也可以通过近似保留解决方案的一些渐近性质。这些类型的方法在文献中通常被认为是渐近保存的。最后，我们心目中的算法必须很少依赖于手头的空间离散化，并且足够简单，以便对数值分析和非线性系统的数学结构知之甚少的用户可以编程。该项目将在很大程度上依赖于pi最近建立的坚实理论基础，并将围绕三个目标进行组织：(i)构建非线性双曲系统的算法，该算法在多项式度，网格结构方面具有鲁棒性，并且具有很少（如果有的话）调谐参数；（ii）构建对混合双曲线与扩散和弥散等其他物理效应的模型具有鲁棒性的近似技术；（iii）构建技术，以保证即使在不完全的物理知识可用的情况下，也能满足离散近似的实际物理边界和热力学不等式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.