On Error Growth Functions
of Runge-Kutta Methods
E. Hairer
and
M. Zennaro
Abstract.
This paper studies estimates of the form $\| y_1 -\widehat y_1 \|
\le \varphi (h\nu )\|y_0-\widehat y_0\|$, where $y_1,\widehat y_1$ are
the numerical solutions of a Runge-Kutta method applied to a stiff
differential equation satisfying a one-sided Lipschitz condition
(with constant $\nu$). An explicit formula for
the optimal function $\varphi (x)$ is given, and it is shown
to be superexponential, i.e., $\varphi (x_1)
\varphi (x_2) \le \varphi (x_1+x_2)$ if $x_1$ and $x_2$ have the same sign.
As a consequence, results on asymptotic stability are obtained.
Furthermore, upper bounds for $\varphi (x)$ are presented that can be easily
computed from the coefficients of the method.
Key Words.
Runge-Kutta methods, error growth functions,
stiff differential equations, B-stability,
asymptotic stability, superexponential functions.