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。该项目将重点关注如何可靠、有效地近似解决一类抽象问题。这些方法将被证明可以产生准确的答案，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，以便他们为这两方面做好准备。工业职业或职业该项目将制定、分析和测试新的窄模板有限差分和不连续伽辽金方法，用于逼近完全非线性偏微分方程的粘度解，例如 Monge-Ampère 方程、Hamilton-Jacobi-Bellman 方程和稳态 Hamilton 方程。 -雅可比方程以及非散度形式的二阶椭圆偏微分方程的解该项目将探索和扩展新的分析技术。 PI 最近开发的目的是证明平稳 Hamilton-Jacobi-Bellman 方程的简单非单调窄模板有限差分方法的可接受性、稳定性和收敛性，另一个目标是基于广义单调性概念来形式化一个抽象收敛框架。比标准单调性更好，因为新方法不需要使用宽模板。新的窄模板方法易于制定和实现，并且具有更高阶。当偏微分方程中存在一阶项时，截断误差优于单调方法。该项目的另一个目标是使用完全非线性的思想来激发新的分析技术来逼近非线性反应扩散方程的正解，这将有助于消除对该奖项反映了 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，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