Runge–Kutta with higher order derivative approximations

✍ Scribed by David Goeken; Olin Johnson

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 168 KB
• Volume: 34
• Category: Article
• ISSN: 0168-9274
• DOI: 10.1016/s0168-9274(99)00128-2

✦ Synopsis

We introduce a form of Runge-Kutta in which it is assumed that the user will evaluate both f and f in solving y = f (x, y) numerically. This allows us to introduce new Runge-Kutta parameters that increase the order of accuracy of the solution with evaluations of f replacing evaluations of f . If f is approximated to sufficient accuracy from past and current evaluations of f , rather than calculated exactly, the order of convergence is retained. The resulting multi-step Runge-Kutta method can be thought of as replacing functional evaluations with approximations of f . Normally, this is an attractive option since f can be approximated to the required accuracy with little arithmetic. Here we present an O(h 3 ) method which requires only two evaluations of f and an O(h 4 ) which requires three.