| Differential equations of the form F (x', x, t) = 0 with Fx' singular arise naturally in many applications and are generally called differential algebraic equations (DAE). There has been an extensive amount of research on numerical solutions of DAEs in recent years. While the classical ODE methods such as backward differentiation and Runga-Kutta methods can be used to numerically solve DAEs, they require the problem to have lower index or special structure.;One approach proposed for solving more general, higher index DAEs is called explicit integration (EI). The original DAE is differentiated a number of times based on certain parameters and the new system of equations is solved using nonlinear least squares methods. The result is a computed ODE whose solutions contain the solutions of the DAE. It is called the least squares completion (LSC). This ODE is then numerically integrated by a classical numerical method.;The EI method is computationally efficient and can be applied to a wide class of DAEs. However, the dynamics of the additional solutions present in the completion can effect the numerical integration, sometimes causing the numerical solutions to move away from the solution manifold. In this thesis, we analyze the additional dynamics of LSCs for linear DAEs. Starting with linear constant coefficient systems, we first examine the structure of the additional dynamics created by the standard LSC and then introduce two methods to modify the completion process so that the LSC will have additional dynamics with desired stability characteristics. The rate of stabilized convergence can be determined a priori by substituting an appropriate value for a parameter. We then extend the results to linear time variable systems. |
