The Runge Kutta methods usually are listed as pairs like 5(4), 3(2), and the like. Do these mean the maximum value of $i$ in $k_i$ in the RK methods. But yet that would be only 1 number. What is the other? And do these pairs occur only as adjacent numbers?
Asked
Active
Viewed 552 times
1 Answers
1
Those are usually used for embedded RK methods, that are methods with two different $b$ vectors with different order. By comparing solutions obtained from different order methods one can estimate local truncation error a posteriori and thus estimate the best integration step to achieve required accuracy.
The order of embedded RK methods is usually written as $p(q)$ with $p$-th order method used for actual computation of $u_{n+1}$ and $q$-th order method is used for the error estimate. $p$ and $q$ may be quite different, but usually they are taken to be close.
For example, famous Dormand-Prince method is a 5(4) order method, since it uses fifth order solution for the next step and the fourth order for the error estimate.
You can find more of this in Solving Ordinary Differential Equations I by Hairer, Norsett and Wanner, sections II.4 and II.5.
uranix
- 7,503
-
Consequently, does Fehlberg 4(5) use a fourth-order solution for the next step, and a fifth-order solution for the error estimate, or is this just an inconsistency? – James Paul Turner Jun 21 '18 at 11:43
-
1@JamesTurner exactly, in RKF4(5) the higher order method is used only for the error estimate, and the lower order is tuned to yield the least truncation error possible. – uranix Jun 22 '18 at 14:23
-
Great, thanks uranix. – James Paul Turner Jun 22 '18 at 14:28