Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Systems with Compact Stencils

本项目的主要目标是系统地发展一类求解对流占优问题的新型高效、高精度Runge-Kutta（RK）间断Galerkin（DG）方法及其应用。新方法具有改进的紧性和局部结构。它们更适合于并行计算和隐式时间推进的计算流体力学模拟。该项目将为对计算数学感兴趣的研究生和/或本科生提供研究机会，并有利于PI部门的课程开发。更详细地说，PI将研究一种新的方法来减少传统RKDG方法的模板大小，该方法通常随着RK阶段的数量而增长。由此产生的新方法被称为紧凑RKDG方法。将从以下几个方面对这些方法进行全面研究。首先，设计求解非线性双曲型守恒律方程的高阶紧致RKDG方法。振荡控制，隐式时间推进和并行计算的技术将被研究。其次，将建立一个严格的理论框架的收敛性，稳定性和误差分析的紧凑RKDG方法。第三，数值技术，以保持解决方案的界限，并探讨其应用于非线性双曲型系统在多维。最后，除了纯粹的对流方程，该方法将扩展到对流扩散问题的模拟粘性flow.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

The main objective of this project is to systematically develop a novel class of efficient and high order accurate Runge–Kutta (RK) discontinuous Galerkin (DG) methods for convection-dominated problems and the related applications. The new methods feature improved compactness and local structures. They are expected to be more suitable for parallel computing and implicit time marching in computational fluid dynamics simulation. They have potential applications in diverse areas such as meteorology, oceanography, gas dynamics, aircraft design, hydraulic engineering, oil recovery simulation, and so on. The project will also provide research opportunities for graduate and/or undergraduate students interested in computational mathematics and benefit curriculum development in the PI’s department. In more detail, the PI will investigate a novel approach to reduce the stencil size of the traditional RKDG methods, which typically grows with the number of RK stages. The resulting new methods are referred to as the compact RKDG methods. A comprehensive study of the methods will be carried out in the following directions. Firstly, high order compact RKDG methods will be designed for nonlinear hyperbolic conservation laws. Techniques for oscillation control, implicit time marching, and parallel computing will be investigated. Secondly, a rigorous theoretical framework for convergence, stability, and error analysis of the compact RKDG methods will be established. Thirdly, numerical techniques to preserve the solution bounds and investigate their applications to nonlinear hyperbolic systems in multidimensions will be developed. Finally, in addition to purely convection equations, the methods will be extended to convection-diffusion problems for simulation of viscous flow.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.