| There has been considerable research on numerical methods for differential algebraic equations (DAE's) of the form {dollar}F(yspprime, y, t){dollar} = 0 where {dollar}Fsb{lcub}yspprime{rcub}{dollar}, is identically singular. Most of the numerical analysis literature on DAE's to date has dealt with DAE's of indices less than three, and often assumes the DAE system to have a special structure. Recently, numerical methods have been proposed that can be used to integrate general nonlinear unstructured higher index DAE's. They are referred to as explicit integration, implicit coordinate partitioning, and coordinate free projection. Previous work on these more general methods has been focused on their theoretical aspects. In this thesis, we examine issues involved with their efficient implementation.; We discuss the use of automatic differentiation in evaluating the functions and Jacobians. For the implicit coordinate partitioning method, we employ a specialized RQ orthogonal decomposition to take the advantages of the structure of the Jacobians. This RQ algorithm leads to a huge savings in computation. For the explicit integration method, We define a special inverse and prove the convergence of the iteration when we use this inverse in the least squares solver. We fully investigate the reuse of Jacobians during the Gauss-Newton iteration. For explicit integration, reuse of Jacobians for more than one time step will lead to different completions during the integration. The analysis and implementations of the integration of these discontinuous completions are carefully examined. We also discuss the termination criteria for the iteration. We show that for implicit coordinate partitioning methods, preserving the constraints takes extra iterations if the partition is not fixed. Numerical results are presented to demonstrate the feasibility and robustness of our speedup techniques. |
