| The terminology differential-algebraic equations (DAEs) comprises systems of differential equations coupled with linear and nonlinear equations. Numerous physical systems, e.g., in multibody dynamics, molecular dynamics, and chemical kinetics, can be modelled according to such formulations.; This thesis deals more specifically with a class of DAEs called mixed index DAEs of index 2 and 3 whose main application domains are given by mechanical systems with holonomic and nonholonomic constraints. For the sake of convenience semi-explicit mixed index DAEs of index 2 and 3 in Hessenberg form are considered. Existence and uniqueness theory is addressed under some specific conditions. Tree structures representing elementary differentials of the exact solution are developed in a systematic fashion.; Specialized Partitioned Additive Runge-Kutta (SPARK) methods are specifically designed for their application to mixed index DAEs of index 2 and 3. Tree structures for the numerical solution are investigated and simplifying assumptions are employed to obtain sharp convergence order estimates of the methods. Relying on the selection of a family of SPARK coefficients, such as Lobatto III A-B-C-C*-D methods, superconvergence can be achieved. Moreover, time-symmetry, reversibility, and symplecticness properties of the flow of the numerical solution hold provided a proper SPARK method is applied. An inexact modified Newton method allows SPARK methods to be implemented efficiently with lower calculation cost.; The dynamic equations in mechanics, which trace the trajectories of mechanical multibody systems, are formulated in general by following the rule for the equations of motions (EOM) of both holonomically and nonholonomically constrained systems, so that SPARK methods for mixed index DAEs of index 2 and 3 play a unified role as general EOM solvers. Throughout various models in mechanics, physical instability caused by index reduction are pointed out and the constraint-preservation property of the SPARK methods are examined. |
