Initial value problems: Numerical methods and mathematics (Q1339319)

The author summarizes his research on Runge-Kutta (RK) and related general linear methods for approximation to the solution of ordinary differential equations. An explanation is given of the way to apply a general series, now called a B-series (Butcher series), to the study of particular RK algorithms. A proof is given of the A-stability of implicit RK algorithms. Criteria for stability and consistency of general linear algorithms are presented. Finally a criterion is given for the stability of a new class of algorithms, called diagonally implicit multistage integration methods (DIMSIM's), developed by the author. Some of these are designed to treat stiff systems by means of parallel computations.