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

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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.