Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems

Abstract

An extensive order analysis is given for the local and global errors of certain classes of Runge–Kutta methods applied to general nonlinear, stiff initial value problems. It turns out that there are three order levels: for linear problems $y' = Ay$ the classical orders apply, for general nonlinear problems, however, two types of order reductions occur. There a distinction has to be made between stiff problems where only $\| {f_y } \|$ is “large” and stiff problems where also some of the other derivatives $f_t ,f_{tt} ,f_{yy} , \cdots $, are “large.” For these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h. The order levels for both classes of problems are considerably lower than the respective conventional orders. These order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems.