ARK method
An
-stage adaptive Runge–Kutta method for the computation of approximations
for the solution
of an initial-value problem
is given by
Here,
is an arbitrary matrix, for stability reasons usually
. For
the method reduces to an explicit Runge–Kutta method. The
are real parameters and
,
, are rational approximations to
for
. The rational matrix functions
,
are defined by
with
and
The computation of
requires the solution of linear systems of algebraic equations only. The coefficients
are determined to give a high order of consistency or B-consistency ([a2]). Applied to the test equation of A-stability,
with
, an adaptive Runge–Kutta method with
yields
By the corresponding choice of stability functions
, adaptive Runge–Kutta methods are A- or L-stable and therefore well suited for stiff systems (cf. Stiff differential system). Furthermore, they can be easily adapted to the numerical solution of partitioned systems, where only a subsystem of dimension
is stiff. Here, by a corresponding choice of
the dimension of the linear systems to be solved can be reduced to
[a1].
References
[a1] | K. Strehmel, R. Weiner, "Partitioned adaptive Runge–Kutta methods and their stability" Numer. Math. , 45 (1984) pp. 283–300 |
[a2] | K. Strehmel, R. Weiner, " -convergence results for linearly implicit one step methods" BIT , 27 (1987) pp. 264–281 |
How to Cite This Entry:
Adaptive Runge–Kutta method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adaptive_Runge%E2%80%93Kutta_method&oldid=15631
This article was adapted from an original article by R. Weiner (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article