Mathematical Sciences: Numerical Methods for Nonlinear Variational Problems

本研究的重点是发展和实现一般和有效的数值算法来解决广泛的非线性变分问题以及非线性特征值问题。两个互补的原则被用于创建新的计算求解器。它们是目标泛函的变换方法和使用匹配的数值技术来解决所产生的问题。所开发的方法将用于许多应用，包括解决磁流体力学中的二维和三维自由边界问题，寻找流体中孤立波理论问题的新解，以及开发具有大量守恒定律的演化问题的新数值技术。

This research is concerned with the development and implementation of general and efficient numerical algorithms for solving a wide range of nonlinear variational problems as well as nonlinear eigenvalue problems. Two complementary principles are utilized for creating new computational solvers. They are the method of transformation of the objective functional and the use of matching numerical techniques for solving the resulting problem. Methods developed will be used in a number of applications which include solving two and three dimensional free boundary problems in magnetohydrodynamics, finding new solutions for problems in the theory of solitary waves in fluids, and developing new numerical techniques for evolution problems with a large number of conservation laws.