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

高阶有限元双曲守恒定律伽辽金逼近的稳定和限制技术

• 批准号: 387630025
• 负责人: Professor Dr. Dmitri Kuzmin
• 金额: --
• 依托单位: Lehrstuhl III: Angewandte Mathematik und Numerik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/387630025?language=en
• 关键词: Stabilization; Limiting; Techniques; Galerkin; Approximations

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.

本计画的目的是进一步发展新的整体凸极限技术，以应用于非线性双曲型问题的有限元素离散化。主要研究者的MCL方法融合了标准的高阶Galerkin离散化和低阶代数版本的Lax-Friedrichs方法，以保证相关最大值原理和熵不等式的有效性。由此产生的非线性计划是可证明的正保持和熵稳定。不涉及任何自由参数和稀疏形式的MCL约束离散化具有紧凑的模板上的分段线性近似的子网格具有相同的节点。第一个供资期的主要重点是使用高阶伯恩斯坦有限元分析和设计连续伽辽金法的代数通量校正工具。拟议的续集项目将扩展的算法框架和理论基础的MCL一般龙格库塔时间离散化，平稳的问题，和不连续的Galerkin（DG）方法。一项特别重要的任务是开发熵校正工具，这些工具不仅适用于纯量非线性半离散问题，而且适用于双曲系统的完全离散逼近。拟议的研究工作还将包括惠普自适应DG计划的发展配备了一种新的通量和斜率限制器的分段线性子细胞近似在非光滑宏细胞。所有新功能都将在开源C++有限元库MFEM（https：//www.example.com）中实现。mfem.org详细的理论研究和其他高分辨率DG计划的比较将进行评估的准确性，鲁棒性和效率的算法。