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力学中以传输为主的广泛系统开发新的计算方法。对于动力学和流体应用，大时间步长的方法仍然不发达。对于量化允许的稳定性的时间步长大小，理论基础尚未建立。在进一步发展移动网格框架的方法和应用于移动边界和界面方面有很大的潜力。该项目将进一步发展最先进的计算工具和理论分析，旨在为目前难以解决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Galerkin(DG)方法，用于线性和非线性传输占优偏微分方程模型。EL-DG方法是线性平流问题的(半拉格朗日)SL-DG方法的推广，其基础是为近似跟踪特征的检验函数设计了一个局部化的伴随问题。这些特征允许灵活性，特别是对于高维和非线性问题，其中的特征很难跟踪。用Runge-Kutta(RK)方法通过直线法对特征逼近过程中产生的误差进行及时积分。这种完全离散的方案被称为“El RK DG”。当特征被很好地逼近时，RK DG框架中非常受限的CFL约束可以被放松，从而导致CPU节省。EL-RK-DG方法可以看作是推广经典欧拉RK-DG公式和SL-DG公式的一般框架。因此，现有关于Eulerian RK DG方法的正性保持限制子、良好平衡处理、渐近保持性质、熵稳定性和误差估计的研究可以潜在地推广到El RK DG框架。一个关键的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照系，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智力优势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.