THE RUNGE-KUTTA THEORY IN A NUTSHELL

摘要

In an earlier paper [Albrecht, SLAM J. Numer. Anal, 24 (1987), pp. 391-406] Runge-Kutta methods were treated as (composite) linear methods. The resulting linear theory is Elementary yielding the order conditions as orthogonal relations from a recursion. This paper further extends this approach and discusses its implications. It is extended to Fehlberg forms, and the global as well as the stage errors am given in great detail, including a new presentation of the principal error function that simplifies error minimization. The orthogonal structure of the order conditions has many advantages; it provides, in particular, a powerful strategy for existence proofs and facilitates the calculation of RK-methods. The recursion, which, originally, was designed to generate the order conditions, is considerably generalized by mappings and becomes a major tool also for the classical theory. it can be used for a recursive generation of rooted bees; also Butcher's elementary differentials and weights as well as his order conditions can be obtained recursively. To this purpose, the elementary weights are reformulated, and the rooted trees are complemented by ''blossoms.'' The whole approach extends to Rosenbrock methods. [References: 8]