High Order Schemes: Robustness, Efficiency, and Stochastic Effects

本项目涉及求解偏微分方程组的高效、高精度数值方法的算法设计和分析。这些方程被用于对航空航天工程、半导体器件设计、天体物理和生物学等不同应用领域中出现的系统进行模拟。即使在当今快速的计算机中，偏微分方程的高效计算解仍然是一个挑战，设计能够在这些应用模型中获得准确解的改进算法是至关重要的。这项研究旨在产生一套强大的计算工具，适合于对这些应用中的复杂解结构进行计算机模拟。该项目通过参与研究为研究生提供培训。该项目研究高阶数值方法的算法开发、分析和应用，包括间断伽辽金(DG)有限元方法和有限体积加权基本无振荡(WENO)格式，用于求解线性和非线性对流占优的偏微分方程组，强调格式的稳健性、效率和随机效应的处理。该项目的重点是算法开发和分析。研究内容包括一类精度越来越高的新的多分辨率WENO格式，求解一般几何问题的笛卡尔网格有限差分格式高阶数值边界条件的反Lax-Wendroff方法，DG格式和其他空间离散的高效稳定的时间推进技术，高精度精度保界格式及其应用，熵稳定的DG方法，DG方法的最优收敛和超收敛分析，随机微分方程组的数值解，以及交通流和空气污染的建模、分析和模拟研究。应用程序激励设计新算法或现有算法中的新功能；将使用数学工具分析这些算法，为它们的适用性和局限性提供指导方针，并提高它们的准确性、稳定性和健壮性；与工程师和其他应用科学家的合作将使这些新算法能够有效应用。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。

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.