Stabilization and Limiting Techniques for Galerkin Approximations of Hyperbolic Conservation Laws With High Order Finite Elements

本项目旨在进一步发展用于非线性双曲型问题有限元离散化的新型整体凸极限（MCL）技术。首席研究员的MCL方法混合了标准的高阶伽辽金离散化和低阶代数版的拉克斯-弗里德里希方法，以保证相关最大原理和熵不等式的有效性。所得到的非线性格式是可证明的保正和熵稳定的。不涉及自由参数，mcl约束离散化的稀疏形式具有相同节点的子网格上分段线性逼近的紧凑模板。第一个资助期的主要重点是分析和设计使用高阶Bernstein有限元的连续Galerkin方法的代数通量校正工具。提出的后续项目将把MCL的算法框架和理论基础扩展到一般龙格-库塔时间离散、平稳问题和不连续伽辽金（DG）方法。一项特别重要的任务将是发展熵校正工具，它不仅适用于标量非线性半离散问题，而且适用于双曲系统的完全离散近似。建议的研究工作还将包括开发配备了一种新的通量和斜率限制器的hp自适应DG方案，用于非光滑宏单元中的分段线性子单元逼近。所有新功能都将在开源c++有限元库MFEM （https://mfem.org）中实现。将进行详细的理论研究并与其他高分辨率DG方案进行比较，以评估所提出算法的准确性、鲁棒性和效率。

This project is aimed at further development of novel monolithic convex limiting (MCL) techniques for finite element discretizations of nonlinear hyperbolic problems. The principal investigator's MCL methodology blends a standard high-order Galerkin discretization and a low-order algebraic version of the Lax-Friedrichs method in a manner which guarantees the validity of relevant maximum principles and entropy inequalities. The resulting nonlinear scheme is provably positivity-preserving and entropy stable. No free parameters are involved and the sparse form of the MCL-constrained discretization has the compact stencil of the piecewise-linear approximation on a submesh with the same nodes. The main focus of the first funding period was on the analysis and design of algebraic flux correction tools for continuous Galerkin methods using high-order Bernstein finite elements. The proposed sequel project will extend the algorithmic framework and theoretical foundations of MCL to general Runge-Kutta time discretizations, stationary problems, and discontinuous Galerkin (DG) methods. A task of particular importance will be the development of entropy correction tools that are suitable not only for scalar nonlinear semi-discrete problems but also for fully discrete approximations to hyperbolic systems. The proposed research endeavors will also include the development of hp-adaptive DG schemes equipped with a new kind of flux and slope limiters for the piecewise-linear subcell approximation in non-smooth macrocells. All new features will be implemented in the open-source C++ finite element library MFEM (https://mfem.org). Detailed theoretical studies and a comparison to other high-resolution DG schemes will be performed to assess the accuracy, robustness, and efficiency of the proposed algorithms.