Potential parallelism in iteration methods

The execution of iteration methods, including iteration methods for time-dependent differential equations, is one of the most time-consuming tasks in scientific computing, especially when large systems have to be solved, large time scales have to be computed or solutions of high quality are required. To decrease the computation time, parallel machines can be used but a detailed analysis of the potential parallelism is needed for an efficient implementation as data dependencies can strongly limit the performance. This article describes and compares the potential parallelism of various iterative solution methods for linear and non-linear equation systems and ordinary differential equations. Methods that have been especially designed for parallel execution as well as parallel implementations of modern efficient sequential solution methods are considered.