Adaptive Runge–Kutta method

ARK method

An $ s $- stage adaptive Runge–Kutta method for the computation of approximations $ u _ {m} $ for the solution $ y ( t _ {m} ) $ of an initial-value problem

$$ y ^ \prime = f ( t,y ) , y ( t _ {0} ) = y _ {0} , t \in [ t _ {0} ,t _ {e} ] , $$

is given by

$$ u _ {m + 1 } ^ {( 1 ) } = u _ {m} , $$

$$ u _ {m + 1 } ^ {( i ) } = R _ {0} ^ {( i ) } ( c _ {i} hT ) u _ {m} + $$

$$ + h \sum _ {j = 1 } ^ { {i } - 1 } A _ {ij } ( hT ) \left [ f ( t _ {m} + c _ {j} h,u _ {m + 1 } ^ {( j ) } ) - Tu _ {m + 1 } ^ {( j ) } \right ] , $$

$$ i = 2 \dots s, $$

$$ u _ {m + 1 } = R _ {0} ^ {( s + 1 ) } ( hT ) u _ {m} + $$

$$ + h \sum _ {j = 1 } ^ { s } B _ {j} ( hT ) \left [ f ( t _ {m} + c _ {j} h,u _ {m + 1 } ^ {( j ) } ) - Tu _ {m + 1 } ^ {( j ) } \right ] . $$

Here, $ T $ is an arbitrary matrix, for stability reasons usually $ T \approx f _ {y} ( t _ {m} ,u _ {m} ) $. For $ T = 0 $ the method reduces to an explicit Runge–Kutta method. The $ c _ {i} $ are real parameters and $ R _ {0} ^ {( i ) } ( z ) $, $ z \in \mathbf C $, are rational approximations to $ e ^ {z} $ for $ z \rightarrow 0 $. The rational matrix functions $ A _ {ij } $, $ B _ {j} $ are defined by

$$ A _ {ij } ( z ) = \sum _ {l = 0 } ^ { \rho _ {i} } R _ {l + 1 } ^ {( i ) } ( c _ {i} z ) c _ {i} ^ {l + 1 } \lambda _ {lj } ^ {( i ) } , $$

$$ B _ {j} ( z ) = \sum _ {l = 0 } ^ { \rho _ {s + 1 } } R _ {l + 1 } ^ {( s + 1 ) } ( z ) \lambda _ {lj } ^ {( s + 1 ) } , $$

with $ \lambda _ {lj } ^ {( i ) } \in \mathbf R $ and

$$ R _ {1} ^ {( i ) } ( z ) = { \frac{R _ {0} ^ {( i ) } ( z ) - 1 }{z} } , $$

$$ R _ {l + 1 } ^ {( i ) } ( z ) = { \frac{lR _ {l} ^ {( i ) } ( z ) - 1 }{z} } . $$

The computation of $ u _ {m + 1 } $ requires the solution of linear systems of algebraic equations only. The coefficients $ \lambda _ {lj } ^ {( i ) } $ are determined to give a high order of consistency or B-consistency ([a2]). Applied to the test equation of A-stability, $ y ^ \prime = \lambda y $ with $ { \mathop{\rm Re} } \lambda \leq 0 $, an adaptive Runge–Kutta method with $ T = \lambda $ yields

$$ u _ {m + 1 } = R _ {0} ^ {( s + 1 ) } ( h \lambda ) u _ {m} . $$

By the corresponding choice of stability functions $ R _ {0} ^ {( s + 1 ) } ( z ) $, 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 $ n _ {s} < n $ is stiff. Here, by a corresponding choice of $ T $ the dimension of the linear systems to be solved can be reduced to $ n _ {s} $[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, "$B$-convergence results for linearly implicit one step methods" BIT , 27 (1987) pp. 264–281