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''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|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 ([[#References|[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|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} $[[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Strehmel,   R. Weiner,   "Partitioned adaptive Runge–Kutta methods and their stability"  ''Numer. Math.'' , '''45'''  (1984)  pp. 283–300</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Strehmel,   R. Weiner,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110320/a11032037.png" />-convergence results for linearly implicit one step methods"  ''BIT'' , '''27'''  (1987)  pp. 264–281</TD></TR></table>