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