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

该项目的主要目标是为计算流体动力学中广泛的运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发达。理论基础尚未建立量化的时间步长大小允许的稳定性。有很大的潜力，在进一步发展的方法，移动网格框架和应用程序的移动边界和接口。该项目将进一步发展最先进的计算工具和理论分析，旨在为目前难以解决的计算模拟提供途径。本项目通过研究对研究生进行培训，开发一类用于线性和非线性输运主导偏微分方程模型的欧拉-拉格朗日（EL）间断Galerkin（DG）方法。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。这样的功能允许灵活性，特别是对于高维和非线性问题，其中的特性是难以跟踪。在逼近特性时出现的误差将通过线法的龙格-库塔（RK）方法在时间上积分。这种完全离散的方案被称为“EL RK DG。“当特征很好地近似时，RK DG框架中非常严格的CFL约束可以放松，从而节省CPU。EL RK DG方法可以被看作是一个通用的框架，推广了经典的欧拉RK DG配方和SL DG配方。因此，现有的研究积极保持限制器，良好的平衡处理，渐近保持性能，熵稳定性和误差估计欧拉RK DG方法可以潜在地推广到EL RK DG框架。一个关键的目标是建立大的时间步长与非线性稳定性。该项目还将探索将EL DG算法推广到移动网格参考系，以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 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.