The numerical solution of transient problems is central to the solution of many scientific and engineering problems. Often the differing time scales make it necessary to solve very large systems of mildly nonlinear equations, which is possible only with methods derived from iterative linear equation solvers. The objective of this research is the application of new iterative method techniques to transient problems, stressing the implementation on parallel computers. Two approaches are considered here, one to solve a set of implicit difference equations that are usually nonlinear, and the other to develop methods of the waveform type, which is iteration applied to a functional formulation of the transient problem. The methods to be employed consist of the analysis and algorithms developed for both initial value problems and for the iterative solution of linear algebraic equations. Large scale problems are of particular interest, and require, especially for 3D simulations, iterative methods on new processors. This work emphasizes the use of currently developed software technology for immediate use in the scientific and engineering communities. The significance will be to improve efficiency of algorithms for solving large scale transient problems on vector and parallel processors.
