The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas (Q1339338)

A study is made of the order of error commited when an implicit Runge- Kutta (RK) algorithm is used together with \(k\)-iterations of Newton's iterative formula in numerical approximation to the solution of an ordinary differential equation. The key to the analysis of the order of error is that the iterated RK method may be viewed as a generalized RK method and the approximation which it generates may be written as a B- series (Butcher series). The order of accuracy is determined both for inverse RK algorithms and for algorithms applicable to differential-algebraic systems of index 1. Results are given for three different iteration schemes: Simple iteration, modified Newton iteration, and full Newton iteration. Results of numerical computations are presented, applied to two examples, in which the observed order of error is compared with the predicted order of error.