High Order Schemes: Bound Preserving, Moving Boundary, Stochastic Effects and Efficient Time Discretization

该项目旨在开发在各种重要的科学和工程应用中求解偏微分方程组的高效和高精度的数值方法，如航空航天工程、半导体器件设计、天体物理和生物应用。即使有了今天速度很快的超级计算机，设计高效可靠的算法以获得这些应用的准确解决方案仍然是必不可少的，在这些应用中，高精度可以提高这些设备的安全性和性能。这些算法将对这些应用中复杂解结构的计算机模拟做出积极的贡献。该项目将包括STEM代表不足群体的学生的劳动力发展。该项目旨在研究算法开发、分析和高阶数值方法的应用，包括不连续伽辽金(DG)有限元方法和有限差分和有限体积加权基本无振荡(WENO)格式。该算法将用于求解线性和非线性对流占优的偏微分方程组，强调保界、移动边界、随机效应和有效的时间离散化。研究课题包括：具有移动边界和界面的数值边界条件的反Lax-Wendroff方法，交通流模拟的前向-后向耦合偏微分方程系统的数学性质和有效的求解器，滞回流动的高阶数值方法，稳健的高阶拉格朗日方法，DG格式和其他空间离散的高效和稳定的时间推进技术，高精度保界格式和应用，包括涉及高度非线性约束和一步Lax-Wendroff时间离散的问题，具有刚性源项的问题，定常双曲型方程和辐射传递方程的高阶DG格式，无振荡的DG方法，以及随机微分方程的数值解。这项研究将为算法的适用性和局限性提供指导，同时提高它们的准确性、稳定性和稳健性。这项研究将包括与工程师和其他应用科学家的合作，以使这些新算法或现有算法中的新功能能够有效应用。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 project aims to develop efficient and high-precision numerical methods for solving partial differential equations in various important scientific and engineering applications, such as aerospace engineering, semiconductor device design, astrophysics, and biological applications. Even with today's fast supercomputers, it is still essential to design efficient and reliable algorithms to obtain accurate solutions to these applications where high precision can improve the safety and performance of those devices. These algorithms will make positive contributions to computer simulations of the complicated solution structure in these applications. The project will include workforce development for students from underrepresented groups in STEM.The project aims to investigate algorithm development, analysis, and application of high-order numerical methods, including discontinuous Galerkin (DG) finite element methods and finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes. The algorithms will be designed to solve linear and nonlinear convection-dominated partial differential equations (PDEs), emphasizing bound preserving, moving boundary, stochastic effects and efficient time discretization. Topics of the research investigations will include an inverse Lax-Wendroff procedure for numerical boundary conditions with moving boundaries and interfaces, mathematical properties and efficient solvers for forward-backward coupled PDE systems from traffic flow modeling, high order numerical methods for hysteretic flows, robust high order Lagrangian methods, efficient and stable time-stepping techniques for DG schemes and other spatial discretizations, high order accurate bound-preserving schemes and applications including problems involving highly nonlinear constraints and one step Lax-Wendroff type time discretizations, problems with stiff source terms, high order DG schemes for stationary hyperbolic equations and radiative transfer equations, oscillation-free DG methods, and numerical solutions of stochastic differential equations. The research will provide guidelines for the algorithms' applicability and limitations while enhancing their accuracy, stability, and robustness. The research will include collaborations with engineers and other applied scientists to enable the efficient application of these new algorithms or new features in existing algorithms.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.