Adaptive Runge–Kutta method

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