On the Contractivity of a Complex Runge-Kutta Scheme

摘要

An attempt to break the order barrier p or = 1 for contractive Runge-Kutta schemes with a special Runge-Kutta scheme of order 2, containing a complex parameter, is made. The implicit Runge-Kutta method is recalled and a previous result is cited, i.e., if the method is contractive with respect to all f belonging to the class of all differential systems (y' - f (t, y), t or = 0, y (0) = y sub 0) with respect to a given norm, where the given norm may be arbitrary, then its order of accuracy p is not 1. Taking a complex backward Euler scheme as an example, the order barrier p or = 1 is found to be in fact unavoidable. This is proven by a counter example.