High Order Schemes: Robustness, Efficiency, and Stochastic Effects

这个项目涉及算法设计和分析的有效，高精度的数值方法求解偏微分方程。这些方程被用于在不同的应用领域，如航空航天工程，半导体器件设计，天体物理学和生物学中产生的系统的模拟。 即使有今天的快速计算机，有效的偏微分方程的计算解决方案仍然是一个挑战，它是必不可少的设计改进的算法，可以用来获得这些应用模型中的准确的解决方案。 该研究旨在产生一套强大的计算工具，适用于这些应用中复杂的解决方案结构的计算机模拟。本项目通过参与研究为一名研究生提供培训。本项目研究高阶数值方法的算法开发、分析和应用，包括间断Galerkin（DG）有限元方法和有限差分和有限体积加权基本无振荡（韦诺）格式，用于求解线性和非线性对流占优偏微分方程，强调格式的鲁棒性，效率和随机效应的处理。 该项目侧重于算法开发和分析。研究课题包括一类精度越来越高的新的多分辨率韦诺格式，求解一般几何问题的笛卡尔网格上有限差分格式的高阶数值边界条件的逆Lax-Wendroff方法，DG格式和其他空间离散的有效和稳定的时间推进技术，高阶精度保界格式及其应用，熵稳定DG方法，DG方法的最优收敛和超收敛分析，随机微分方程的数值解，以及交通流和空气污染的建模、分析和仿真研究。 应用激发了新算法或现有算法中新功能的设计;数学工具将用于分析这些算法，为其适用性和局限性提供指导，并提高其准确性，稳定性和鲁棒性;与工程师和其他应用科学家的合作将使这些新算法的有效应用成为可能。该奖项反映了NSF的法定使命，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来提供支持。

This project concerns algorithm design and analysis of efficient, highly accurate numerical methods for solving partial differential equations. Such equations are used in simulation of systems arising in diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biology. Even with today's fast computers, efficient computational solution of partial differential equations remains a challenge, and it is essential to design improved algorithms that can be used to obtain accurate solutions in these application models. The research aims to produce a suite of powerful computational tools suitable for computer simulations of the complicated solution structure in these applications. The project provides training for a graduate student through involvement in the research.This project conducts research in 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, for solving linear and nonlinear convection-dominated partial differential equations, emphasizing scheme robustness, efficiency, and the treatment of stochastic effects. The project focuses on algorithm development and analysis. Topics of investigation include a new class of multi-resolution WENO schemes with increasingly higher order of accuracy, an inverse Lax-Wendroff procedure for high-order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, efficient and stable time-stepping techniques for DG schemes and other spatial discretizations, high order accurate bound-preserving schemes and applications, entropy stable DG methods, optimal convergence and superconvergence analysis of DG methods, numerical solutions of stochastic differential equations, and the study of modeling, analysis, and simulation for traffic flow and air pollution. Applications motivate the design of new algorithms or new features in existing algorithms; mathematical tools will be used to analyze these algorithms to give guidelines for their applicability and limitations and to enhance their accuracy, stability, and robustness; and collaborations with engineers and other applied scientists will enable the efficient application of these new 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.