Mathematical Sciences: Computational Methods for Multiconstrained Nonlinear Variational Problems

所提出的研究集中在一种新的计算方法上，该方法被用来求解具有大的、可能是无限的约束集的变分问题。本文从误差分析、离散化技术、泛化和收敛加速等方面对该方法进行了研究。在这种方法的基础上，有望为一类平衡和慢演化问题以及具有大量守恒律的演化问题开发计算算法。在这些计算算法的帮助下，提出了一大类在物理和工程中的应用。应用问题包括理想磁流体力学(托卡马克设计、实验数据分析)、流体力学(设计具有更好的收敛和稳定性的高效计算算法)和天体物理学。提出了求非线性波动方程周期解和二阶动力系统拟周期解的数值方法的变分方法。

The proposed research is focused on a new computational method developed for solving variational problems with a large, possibly infinite, set of constraints. An investigation of various aspects of this method is proposed, with emphasis on the error analysis, discretization techniques, generalizations and convergence acceleration. On the basis of this method computational algorithms are expected to be developed for a class of equilibrium and slow evolution problems and for evolution problems with a large number of conservation laws. A large class of applications in physics and engineering is proposed for the investigation with the help of these computational algorithms. Applied problems include problems in ideal magnetohydrodynamics (tokamak design, analysis of the experimental data), fluid dynamics (design of efficient computational algorithms with improved convergence and stability properties) and astrophysics. The variational approach is also proposed for developing numerical methods for finding periodic solutions of nonlinear wave equation and quasi-periodic solutions of second order dynamical systems.