Narrow-Stencil Numerical Methods for Approximating Nonlinear Elliptic Partial Differential Equations

逼近非线性椭圆偏微分方程的窄模板数值方法

• 批准号: 2111059
• 负责人: Thomas Lewis
• 金额: $ 12万
• 依托单位: University of North Carolina Greensboro
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111059&HistoricalAwards=false
• 关键词: Narrow; Stencil; Numerical; Methods; Approximating

The project will develop new computational methods for simulating various applications in astrophysics, fluid mechanics, image processing, wave propagation, geometric optics, biology, and combustion theory. The project will focus on how to reliably and efficiently approximate solutions to a class of abstract problems that can be used to model various phenomena relevant to the applications. The methods will be proven to yield accurate answers and will also be simple to implement. The project will involve activities towards mentoring and broadly training graduate students so that they are prepared for both an industrial career or a career in academia. The project will formulate, analyze, and test new narrow-stencil finite difference and discontinuous Galerkin methods for approximating viscosity solutions of fully nonlinear PDEs such as the Monge-Ampère equation, the Hamilton-Jacobi-Bellman equation, and the stationary Hamilton-Jacobi equation as well as solutions of second order elliptic PDEs in non-divergence form. The project will explore and extend the novel analytic techniques the PI recently developed to prove the admissibility, stability, and convergence of a simple non-monotone narrow-stencil finite difference method for stationary Hamilton-Jacobi-Bellman equations. Another objective is to formalize an abstract convergence framework based on the notion of generalized monotonicity rather than standard monotonicity, as the new methods do not require the use of wide-stencils. The new narrow-stencil methods are easy to formulate and implement and have higher-order truncation errors than monotone methods when first-order terms are present in the PDE. Another goal of the project is to use fully nonlinear ideas to motivate new analytic techniques for approximating positive solutions of nonlinear reaction diffusion equations; these will help eliminate the need for a comparison principle assumption when approximating fully nonlinear problems.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.

该项目将开发新的计算方法，用于模拟天体物理学、流体力学、图像处理、波传播、几何光学、生物学和燃烧理论中的各种应用。 该项目将专注于如何可靠和有效地近似解决一类抽象问题，可用于模拟与应用程序相关的各种现象。 这些方法将被证明能够产生准确的答案，并且易于实施。 该项目将涉及指导和广泛培训研究生的活动，使他们为工业生涯或学术生涯做好准备。 该项目将制定，分析和测试新的窄模板有限差分和不连续Galerkin方法，用于近似完全非线性偏微分方程的粘性解，如Monge-Ampère方程，Hamilton-Jacobi-Bellman方程和稳态Hamilton-Jacobi方程以及非发散形式的二阶椭圆偏微分方程的解。 该项目将探索和扩展PI最近开发的新的分析技术，以证明一个简单的非单调窄模板有限差分方法的容许性，稳定性和收敛性，用于固定的Hamilton-Jacobi-Bellman方程。 另一个目标是形式化一个抽象的收敛框架的基础上的广义单调性的概念，而不是标准的单调性，因为新的方法不需要使用宽的stenosis。 新的窄模板方法是很容易制定和实施，并有高阶截断误差比单调方法时，一阶项存在于偏微分方程。 该项目的另一个目标是使用完全非线性的想法来激励新的分析技术，以近似非线性反应扩散方程的正解;这将有助于消除在近似完全非线性问题时对比较原理假设的需要。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

项目成果

A NARROW-STENCIL FRAMEWORK FOR CONVERGENT NUMERICAL APPROXIMATIONS OF FULLY NONLINEAR SECOND ORDER PDES

全非线性二阶偏微分方程收敛数值逼近的窄模板框架

• 发表时间: 2022
• 期刊: Electronic journal of differential equations
• 影响因子: 0.7
• 作者: XIAOBING FENG, THOMAS LEWIS

Convergence, stability analysis, and solvers for approximating sublinear positone and semipositone boundary value problems using finite difference methods

• DOI: 10.1016/j.cam.2021.113880
• 发表时间: 2021-10
• 期刊: J. Comput. Appl. Math.
• 作者: T. Lewis;Q. Morris;Yi Zhang

Penalty parameter and dual-wind discontinuous Galerkin approximation methods for elliptic second order PDEs

• DOI: 10.58997/ejde.conf.26.l1
• 发表时间: 2022-08
• 期刊: Electronic Journal of Differential Equations
• 影响因子: 0.7
• 作者: T. Lewis;Aaron Rapp;Yi Zhang

Consistency results for the dual-wind discontinuous Galerkin method

双风间断伽辽金法的一致性结果

• DOI: 10.1016/j.cam.2023.115257
• 发表时间: 2023
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Lewis, Tom;Rapp, Aaron;Zhang, Yi
• 通讯作者: Zhang, Yi

Convergence analysis of a symmetric dual-wind discontinuous Galerkin method for a parabolic variational inequality

抛物型变分不等式的对称双风间断伽辽金法的收敛性分析

• DOI: 10.1016/j.cam.2022.114922
• 发表时间: 2023
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Boyana, Satyajith Bommana;Lewis, Thomas;Rapp, Aaron;Zhang, Yi
• 通讯作者: Zhang, Yi