Cash–Karp method

From Wikipedia, the free encyclopedia

In numerical analysis, Cash–Karp is a method for solving ordinary differential equations (ODEs). The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Dormand–Prince (RKDP).

The Butcher tableau is:

	0
	1/5	1/5
	3/10	3/40	9/40
	3/5	3/10	−9/10	6/5
	1	−11/54	5/2	−70/27	35/27
	7/8	1631/55296	175/512	575/13824	44275/110592	253/4096
		37/378	0	250/621	125/594	0	512/1771
		2825/27648	0	18575/48384	13525/55296	277/14336	1/4

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

[edit] References

• J. R. Cash, A. H. Karp. "A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides", ACM Transactions on Mathematical Software 16: 201-222, 1990. doi:10.1145/79505.79507.