Mathematical Sciences: A General Approach for the Numerical Solution of DAEs

微分代数方程(DAE)是形式为F(y，y，u(T)，t)=0的隐式常微分方程，其中y(T)是向量值函数，Jacobian Fy是奇异矩阵，u(T)是源项或控制项的向量值函数。许多问题最初最自然地被建模为DAE。例如，它们出现在飞行控制、机器人、化学过程控制以及力学中的接触或约束问题中的各种控制问题中。此外，在用直线法求解偏微分方程组时，以及在许多电路模型中，都会出现微分代数方程。DAE有时也称为单系统或描述符系统。拟议研究的长期目标是让科学家和工程师能够直接使用原始的隐式模型，从而减少设计和模拟时间，并有助于考虑更复杂的问题。现有的数字代码虽然非常有用，但需要特殊的问题结构，并且不能处理许多感兴趣的问题。该项目将开发一种求解非线性微分代数方程的通用数值方法。当存在显式模型时，这些结果也将是有意义的，但稀疏性、非线性或对方程参数的依赖性使得原始隐式系统优于低阶但非稀疏的(或难以计算局部的)显式系统。

A differential algebraic equation (DAE) is an implicit ordinary differential equation of the form F(y,y,u(t),t)=0 where y(t) is a vector valued function, the Jacobian F y is a singular matrix, and u(t) is a vector valued function of source or control terms. Many problems are most naturally initially modeled as a DAE. They arise, for example, in a variety of control problems in flight control, robotics, chemical process control, and contact or constrained problems in mechanics. In addition, differential algebraic equations occur in the solution of partial differential equations by the method of lines and in many circuit models. DAEs are sometimes also called singular or descriptor systems. The long range goal of the proposed research is for scientists and engineers to be able to work directly with the original implicit model thus reducing design and simulation time as well as facilitating the consideration of more complex problems. Existing numerical codes, while very useful, require special problem structure and do not handle many problems of interest. This project will develop a general numerical method for nonlinear differential algebraic equations. These results will also be of interest when explicit models exist, but the sparsity, nonlinearity, or dependence on parameters of the equations, make the original implicit system preferable to a lower order, but non-sparse, (or difficult to compute local) explicit system.