The work presented in this report deals with the development of a fast and fairly accurate Computer-Aided Design software for simulating very-large-scale-integrated VLSI circuits. The methods rely on piecewise linearized nonlinear elements in the circuits. The piecewise linear approaches explored in this work are 1 a fast piecewise linear Gauss-Seidel waveform relaxation method 2 a slower but more accurate piecewise linear method based on simplices and 3 a Gauss-Seidel piecewise linear method with dynamic partitioning. Also described is a mixed method which combines the fast piecewise linear method and the dynamic partitioning method. The circuit to be analyzed is partitioned into dc-connected subcircuits and then sequenced for analysis. Small subcircuits are solved using the fast piecewise linear method while large subcircuits, including the strongly connected components in the circuit, are solved using the dynamic partitioning method. A parallel implementation of the Gauss-Seidel piecewise linear method with dynamic partitioning on a uniprocessor computer is studied. Algorithms for the parallel implementation of the dynamic partitioning approach on a multiprocessor with shared memory Alliant FX8 are also explained in detail. The piecewise linear methods presented in this work have been implemented in a set of programs called PLATINUM. The waveforms generated by PLATINUM are fairly accurate as compared to those for SPICE2, and the speedup for a uniprocessor machine is over two orders of magnitude, while the parallel implementation gives an additional 4 to 6 times speed improvements.