Numerical Methods and Algorithms for Fully Nonlinear Second Order Evolution Equations with Applications

全非线性二阶演化方程的数值方法和算法及其应用

• 批准号: 1016173
• 负责人: Xiaobing Feng
• 金额: $ 22.5万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016173&HistoricalAwards=false
• 关键词: Numerical; Methods; Algorithms; Fully; Nonlinear

Fully nonlinear second order partial differential equations (PDEs) are referred to a class of nonlinear second order PDEs which are nonlinear in (at least one) second order partial derivatives of unknown functions. Such a class of PDEs arise from many scientific and engineering fields including astrophysics, differential geometry, geostrophic fluid dynamics, image processing, kinetic theory, materials science, mass transportation, meteorology, and optimal control. They constitute the most difficult class of PDEs to analyze analytically and to approximate numerically. Building on the PI's recent success on developing convergent and efficient numerical methods and algorithms for fully nonlinear second order (time-independent) elliptic PDEs, the proposed research project intends to carry out a comprehensive and systematic study of numerical methods and algorithms for fully nonlinear second order (time-dependent) evolution PDEs. The objectives of the proposed research include (i) to develop the vanishing moment method and the moment solution theory for fully nonlinear second order evolution PDEs, (ii) to develop fully discrete Galerkin type numerical methods (e.g. finite element methods, mixed finite element methods, spectral and discontinuous Galerkin methods) for fully nonlinear second order evolution PDEs based on the vanishing moment methodology, (iii) to apply and/or to adapt the developed numerical methods to a number of emerging application problems which are governed by fully nonlinear second order PDEs, (iv) to develop computer codes for implementing the proposed numerical methods. As numerical approximations of fully nonlinear second order evolution PDEs is an untouched sub-area within the numerical PDEs and those PDEs arise from many important applications in astrophysics, differential geometry, geostrophic fluid dynamics, image processing, kinetic theory, materials science, mass transportation, meteorology, and optimal control, the completion of the proposed research project is expected to have a profound impact on solving this class of PDEs and on providing the much needed capability and enabling tools for solving a range of important application problems which are governed by fully nonlinear second order PDEs. As a by-product, the moment solution theory is expected to give some insights to our understanding of the viscosity solution theory, and might be very likely to provide a logical and natural generalization and extension for the viscosity solution theory which is not natural and neither practical from the computational point of view. The educational component of this project is to engage and train graduate students in developing necessary applied and computational mathematics knowledge and skills so that they can pursue a successful career in science and engineering in the future.

全非线性二阶偏微分方程（PDE）是指一类非线性二阶偏微分方程，其在未知函数的（至少一个）二阶偏导数上呈非线性。此类偏微分方程源自许多科学和工程领域，包括天体物理学、微分几何、地转流体动力学、图像处理、动力学理论、材料科学、大众运输、气象学和最优控制。它们构成了最难进行分析分析和数值近似的偏微分方程。基于PI最近在开发全非线性二阶（与时间无关）椭圆偏微分方程的收敛和高效数值方法和算法方面取得的成功，拟议的研究项目旨在对全非线性二阶（与时间无关）椭圆偏微分方程的数值方法和算法进行全面和系统的研究阶次（时间相关）演化偏微分方程。本研究的目标包括（i）开发完全非线性二阶演化偏微分方程的消失矩法和矩解理论，（ii）开发完全离散伽辽金型数值方法（例如有限元法、混合有限元法） ，谱和不连续伽辽金方法）用于基于消失矩方法的完全非线性二阶演化偏微分方程，（iii）应用和/或使开发的数值方法适应许多由完全非线性二阶控制的新兴应用问题偏微分方程，(iv) 开发计算机代码以实现所提出的数值方法。 由于完全非线性二阶演化偏微分方程的数值近似是数值偏微分方程中未触及的子区域，并且这些偏微分方程源自天体物理学、微分几何、地转流体动力学、图像处理、动力学理论、材料科学、大众运输等领域的许多重要应用，气象学和最优控制，拟议研究项目的完成预计将对解决此类偏微分方程产生深远的影响，并为解决一系列重要的应用问题提供急需的能力和支持工具，这些问题包括：由完全非线性二阶偏微分方程控制。作为副产品，矩解理论有望为我们对粘度解理论的理解提供一些见解，并且很可能为粘度解理论提供逻辑和自然的概括和扩展，而这既不是自然的，也不是从计算的角度来看很实用。该项目的教育部分是吸引和培训研究生发展必要的应用和计算数学知识和技能，以便他们将来能够在科学和工程领域取得成功的职业生涯。