Eulerian-Lagrangian Runge-Kutta Discontinuous Galerkin Methods for Nonlinear Kinetics and Fluid Models

非线性动力学和流体模型的欧拉-拉格朗日龙格-库塔不连续伽辽金方法

基本信息

批准号：
2111253
负责人：
Jing-Mei Qiu
金额：
$ 30.46万
依托单位：
University of Delaware
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111253&HistoricalAwards=false
关键词：
Eulerian Lagrangian Runge Kutta Discontinuous

项目摘要

The primary goal of the project is the development of new computational methodologies for a wide range of transport-dominant systems in computational fluid dynamics. Methods with large time step sizes are still underdeveloped for kinetic and fluid applications. Theoretical foundations are yet to be established for quantifying the time stepping sizes allowed for stability. There is great potential in further development of methodology for a moving mesh frame and application to moving boundaries and interfaces. This project will further state-of-art computational tools and theoretical analysis and aims to provide avenues for computational simulations that are currently intractable. The project involves training of graduate students through involvement in the research.This project will develop a class of Eulerian-Lagrangian (EL) Discontinuous Galerkin (DG) approaches for linear and nonlinear transport-dominant partial differential equation models. The EL DG method is a generalization of the (semi-Lagrangian) SL DG method for linear advection problems, based on the design of a localized adjoint problem for the test function that approximately tracks characteristics. Such features allow flexibility, especially for high dimensional and nonlinear problems, where characteristics are difficult to track. The errors occurred in approximating characteristics will be integrated in time by Runge-Kutta (RK) methods via the method-of-lines approach. This fully discrete scheme is termed "EL RK DG." When the characteristics are approximated well, the very restrictive CFL constraint in the RK DG framework can be relaxed, leading to CPU savings. The EL RK DG method can be viewed as a general framework generalizing both the classical Eulerian RK DG formulation and the SL DG formulation. Thus, existing research on positivity preserving limiters, well-balanced treatments, asymptotic preserving properties, entropy stability, and error estimates on Eulerian RK DG methods can be potentially generalized to the EL RK DG framework. A key goal is to establish large time-stepping size with nonlinear stability. The project will also explore generalization of the EL DG algorithm to a moving mesh reference frame for tracking material interfaces and moving boundaries.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.

该项目的主要目标是为计算流体动力学中的各种传输主导系统开发新的计算方法。对于动力学和流体应用，具有大时间步长的方法仍然不完善。用于量化实现稳定性的时间步长大小的理论基础尚未建立。进一步开发移动网格框架的方法及其在移动边界和界面上的应用具有巨大潜力。该项目将进一步发展最先进的计算工具和理论分析，旨在为目前难以处理的计算模拟提供途径。该项目包括通过参与研究来培训研究生。该项目将开发一类欧拉-拉格朗日 (EL) 间断伽辽金 (DG) 方法，用于线性和非线性输运主导的偏微分方程模型。 EL DG 方法是线性平流问题的（半拉格朗日）SL DG 方法的推广，基于近似跟踪特性的测试函数的局部伴随问题的设计。这些功能提供了灵活性，特别是对于特征难以跟踪的高维和非线性问题。近似特性中出现的误差将通过龙格-库塔（RK）方法通过线法方法及时积分。这种完全离散的方案称为“EL RK DG”。当特性近似良好时，可以放宽 RK DG 框架中非常严格的 CFL 约束，从而节省 CPU。 EL RK DG 方法可以被视为概括了经典欧拉 RK DG 公式和 SL DG 公式的通用框架。因此，关于欧拉 RK DG 方法的正性保持限制器、均衡处理、渐近保持特性、熵稳定性和误差估计的现有研究可以潜在地推广到 EL RK DG 框架。一个关键目标是建立具有非线性稳定性的大时间步长。该项目还将探索将 EL DG 算法推广到移动网格参考系，用于跟踪材料界面和移动边界。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。