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

完全非线性二阶偏微分方程指的是一类非线性的二阶偏微分方程组，它们是未知函数的(至少一个)二阶偏导数的非线性。这类偏微分方程组产生于天体物理、微分几何、地转流体动力学、图像处理、动力学理论、材料科学、质量输运、气象学和最优控制等众多科学和工程领域。它们构成了一类最难分析和数值逼近的偏微分方程组。在PI最近成功地发展了收敛且高效的完全非线性二阶(时无关)椭圆型方程数值方法和算法的基础上，本研究项目旨在对完全非线性二阶(时相关)发展偏微分方程组的数值方法和算法进行全面系统的研究。所提出的研究目标包括(I)发展完全非线性二阶发展偏微分方程组的消失矩方法和矩解理论，(Ii)发展基于消失矩方法的全离散Galerkin型数值方法(例如有限元方法、混合有限元方法、谱方法和间断Galerkin方法)用于完全非线性二阶发展方程组，(Iii)应用和/或使所发展的数值方法适用于由完全非线性二阶偏微分方程组控制的许多新兴应用问题，(4)开发用于实施拟议的数值方法的计算机代码。由于完全非线性二阶发展偏微分方程组的数值近似是数值偏微分方程组中尚未触及的一个子领域，而这些偏微分方程组在天体物理、微分几何、地转流体动力学、图像处理、动力学理论、材料科学、质量输运、气象学和最优控制等领域中有着重要的应用，因此，该研究项目的完成将对解决这类偏微分方程组以及为解决由完全非线性二阶偏微分方程组所支配的一系列重要应用问题提供迫切需要的能力和使能工具产生深远影响。作为一种副产品，矩理论有望给我们对粘性解理论的理解提供一些启示，并很有可能为从计算角度看既不自然又不实用的粘性解理论提供一个合乎逻辑和自然的推广。这个项目的教育部分是让研究生参与并培训他们发展必要的应用和计算数学知识和技能，以便他们能够在未来的科学和工程领域取得成功。

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.