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

本计画的目标是在一、二、三个空间维度的非结构化网格上，建构非线性守恒系统的强健逼近技术。这类问题涉及工程学的许多领域（机械、航空航天、核能、海洋等）。和科学（天体物理学，天体物理学）。一套新的鲁棒近似技术，用于解决复杂的非线性守恒方程在现实环境中，也将有利于许多应用程序，涉及模型混合双曲性与其他物理效应，如扩散和色散。该项目的成果将通过研究生课程、学生辅导、研讨会、会议演示、出版物以及与美国各机构同事的直接合作进行传播。在这个项目中开发的材料将被纳入一个先进的类守恒方程，每两年在数学系在得克萨斯州A M.在这个项目中，鲁棒性意味着所提出的方法保证提供解决方案，满足物理和物理约束（一般基于准凹或凹泛函）。鲁棒性还意味着，即使网格不够精细到处于某个渐近范围内（即，必须避免锁定）。这些类型的方法在文献中通常被认为是渐近保持的。最后，我们所考虑的算法必须很少依赖于手头的空间离散化，并且足够简单，以便在数值分析和非线性系统的数学结构方面知之甚少的用户进行编程。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，并将围绕三个目标组织：（i）构建非线性双曲系统的算法，这些算法在多项式次数，网格结构方面是鲁棒的，并且具有很少的（如果有的话）调谐参数;（ii）构建近似技术，该技术对于将双曲性与扩散和分散等其他物理效应混合的模型是稳健的;（三）该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

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.