Solving index and near index problems in dynamic simulation

This thesis presents a numerical algorithm to detect and solve mixed systems of nonlinear ordinary differential and algebraic equations which have an index problem. Exact numerical differentiation is suggested to generate the equations needed; symbolic elimination is not required as in many other algorithms. The so-called initialization and propagation problems are shown to be the same; they can therefore be handled by a common approach. Numerical as well as structural singularities arising in index problems are accommodated. In principle the algorithm can be used with any integration scheme, including an explicit one.; The algorithm for solving nonlinear index problems is extended to include stiff DAE systems. Stiff DAE systems are shown to be near index problems. Near high index problems cannot be solved by existing stiff ODE/DAE solvers, such as LSODI. Stiff problems should be classified by index. Solutions to these problems can be expressed and controlled in terms of polynomials in the small parameters which are responsible for the near singularity at the solution point.; The numerical algorithm presented in this thesis is suitable for solving index problems which previously could not be solved. The algorithm can also solve stiff DAE systems accurately and stably even with an explicit integration method.; This thesis also addresses implementation of the algorithm. A numerical singularity checking method is suggested to detect the singularity or the near singularity of a matrix and to identify the equations responsible for the singularity or the near singularity. An exact numerical differentiation method is also suggested to generate elements of the Jacobian matrix and the equations needed.