This dissertation presents numerical methods for solving two classes of or-dinary diferential equations (ODE) based on single-step integration meth-ods. The first class of equations addressed describes the mechanical dynamics of constrained multibody systems. These equations are ordinary differential equations (ODE) subject to algebraic constraints. Accordinly they are called differential-algebraic equations (DAE).

Specific contributions made in this area include an explicit transforma-tion between the Hessenberg index-3 form for constrained mechanical systems to a canonical state-space form used in the nonlinear control communities. A hybrid solution method was developed that incorporates both sliding-mode control (SMC) from the controls literature and post-stabilization from the DAE related literature. The process of developing the hybrid method produced insights into both areas in a way that allowed both areas to benefit from the other’s strengths. First, the hybrid method produced an accurate and efficient method for simulating sliding-mode control systems. A technique called post-stabilization provides a more efficient method for simulating SMC systems than conventional methods using the discontinuous control term. Sec-ond, use of SMC mathematical framework allows the hybrid method to handle arbitrary, or inconsistent initial conditions.

The second class of equations addressed here are discontinuous ODE. Specific contributions made in solving DODE include further classification of discontinnuities into parametric or structural discontinuities as well as unilateral or bilateral events. Consistent event location and discontinuity sticking from Park and Barton[56] originally addressed bilateral events only and were implemented in a single-step environment and then extended to address uni-lateral events as well. An effective detection scheme was developed using low-order interpolants for detecting most events in the correct order. For rare cases when the detection scheme fails, a try-catch model was implemented to deal with two possible failure scenarios. The detection and location methods successfully handled all events in the correct order for the benchmark problems solved. Lastly, a region of concurrency was developed that can provide large efficiency gains for some systems containing multiple closely spaced events.