Comparing the accuracy of RK methods: A rigorous approach

Ander Murua, Joseba Makazaga

KZAA saila, Euskal Herriko Unibertsitatea, 649 PK., 20080 Donostia - San Sebastian, Spain

ander@si.ehu.es

Poster

We present a general procedure to obtain rigorous estimates for the local error of Runge-Kutta methods when applied to an ODE system $\dot y = f(y)$. Under the assumption that $f(y)$ is real analytic in an open set of phase space (this assumption can somewhat be relaxed), we give a procedure to obtain, for each particular $p$th order Runge-Kutta method, an estimate for the local error $\delta(y,h)$ of the form

$$\begin{eqnarray*} \vert\vert\delta(y,h)\vert\vert \leq h \, M \, d(h L), \quad d... ...) := \sum_{j=p+1}^{N-1} \tau^{j-1} c_j + \tau^{N-1} r_{N}(\tau), \end{eqnarray*}$$

where $M$ and $L$ are constants that only depend on $f$ and $y$, and the function $d(\tau)$ (constants $c_j$ and the remainder function $r_N(\tau)$) only depend on the Butcher tableau of the RK scheme. The norm $\vert\vert\cdot\vert\vert$ can be arbitrarily chosen, and only $M$ depends on that choice. The remainder function $r_{N}(\tau)$ is defined for $0\leq \tau \leq \kappa$ where $\kappa>0$ is a constant that depends on the Butcher tableau. The local error can thus be estimated for all $h\leq \kappa/L$.

We propose to compare the theoretical accuracy of different RK schemes by comparing the corresponding function $d(\tau)$. We claim that $d(\tau)$ is a fair indicator of the average behaviour of the local error performed by the RK scheme when applied to general ODEs satisfying our assumptions. Numerical experiments are presented to support this claim.

We use our theory to evaluate the theoretical efficiency of different RK schemes (of possibly different orders) with the same number of effective stages by comparing the diagrams of their corresponding functions $d(\tau)$. We use this approach to chose, among several Runge-Kutta methods, the most efficient schemes for different ranges of tolerances.