Abstracts Keynote talks

A Runge–Kutta method takes small time steps, to approximate the solution to an initial value problem. How accurate is this approximation? If the error is asymptotically proportional to hp, where h is the stepsize, the Runge–Kutta method is said to have “order” p. To find p, write the exact solution, after a single time-step, as a Taylor series, and compare with the Taylor series for the approximation. This seems to be the answer to finding numerical orders – it is the truth pure and simple. But, this answer is too glib; “The truth is rarely pure and never simple”. Perhaps surprisingly, the order of a numerical method, for a scalar problem y = f (x,y), can be greater than when the same method is applied to a high-dimensional problem y = f (y). A consideration of the group structure of Runge–Kutta methods, leads to a more general definition of order. Methods that possess this enhanced “effective order”, lead to genuinely improved numerical performance. Furthermore, the Hopf Algebra of rooted trees, which is at the heart of Runge–Kutta theory, turns out to be the appropriate basis for the order analysis of multi-stage multi-value numerical methods.