1. T. Feagin University of Houston – Clear Lake Houston, Texas, USA June 24, 2009 1 High-Order Explicit Runge-Kutta Methods Using m-Symmetry

2. Background and introduction The Runge-Kutta equations of condition New variables Reformulated equations m-symmetry Finding an m-symmetric method Numerical experiments 2

3. 3

4. 4 h - the stepsize t0t0 t 0 + h where

5. 5 The order of the formula m The number of new equations of order m The total number of equations for order m Number of stages n The number of unknowns n (n+1)/2 11111 21223 32436 448410 5917621 62037728 74885945 81152001166 928648615120 10719120517153 1118423047-- 124766781325325 131248620299-- 14329735327235630 1587811141083

6. 6 for

7. 7 wherefor

8. 8 wherefor one of the column simplifying assumptions when zero one of the row simplifying assumptions when zero

9. 9 for all other values ofin the range for

10. 10 The set of integer subscripts can be partitioned into three subsets quadrature points non-matching points matching points Q M N Theorem: Any m-symmetric Runge-Kutta method is of order m.

11. 11 quadrature pointsQ for 0 12 13 14 15 16 24

12. 12 for 1 7 4 2 6 9 10 23 19 21 22 20 18 17 matching pointsM whereand is the smallest value ofsuch that

13. 13 11 8 3 5 non-matching pointsN for

14. 14

15. 15 Determine a quadrature formula of order m or higher with u weights and u nodes Gauss-Lobatto formulae are a possible and usually convenient choice Determine (or establish equations governing the values of) the points leading up to α k2 (the first internal quadrature point) such that the order at the quadrature points is m/2

16. 16 Identify the matching and non-matching points Obtain values for any of the α k ‘s yet to be determined (i.e., solve nonlinear equations) Select non-zero values for the free parameters (c k ‘s at the matching points) such that, … Solve the remaining equations from the definition to make the method m-symmetric

17. 17 p k,6,21 vs k r k4 vs k Example plots for the 12 th -order method

18. Seeking to reduce the local truncation errors by minimizing size and number of the unsatisfied 13 th -order terms (more than 92% are satisfied) Trying to keep the largest coefficient (in absolute value) to a reasonable level (~12) Trying to maintain a reasonably large absolute stability region 18 Re(hλ) Im(hλ)

19. 19 RK12 RK10H RK8CV RK6B RK4 -log10(error) log10(NF) Eccentricity = 0.4 Fixed step integration

20. 20 The true error and the estimated error for RK12(10)

21. 21

22. 22 Variable step RK12(10) Pleiades problem GBS

23. 23

24. 24 Kepler Problem (e = 0.1)

25. 25

26. 26 Kepler Problem (e = 0.9)

27. 27 W. B. Gragg, On extrapolation algorithms for ordinary initial value problems. SIAM J. Num. Anal., 2 (1965) pp. 384-403 T. Feagin,, A Tenth-Order Runge-Kutta Method with Error Estimate, Proceedings (Edited Version) of the International MultiConference of Engineers and Computer Scientists 2007, Hong Kong E. Hairer, A Runge-Kutta method of order 10, J. Inst. Math. Applics. 21 (1978) pp. 47-59 E. Fehlberg, Classical Fifth-, Sixth-, Seventh-, and Eighth-Order Runge-Kutta Formulas with Stepsize Control, NASA TR R-287, (1968) E. Baylis Shanks, Solutions of Differential Equations by Evaluations of Functions, Math. Comp. 20, No. 93 (1966), pp. 21-38 P.J. Prince and J.R. Dormand, High-order embedded Runge-Kutta formulae, J Comput. Appl. Math., 7 (1981), pp. 67-76 J.H. Verner, The derivation of high order Runge Kutta methods, Univ. of Auckland, New Zealand, Report No. 93, (1976) Hiroshi Ono, On the 25 stage 12th order explicit Runge--Kutta method, JSIAM Journal, Vol. 16, No. 3, 2006, p. 177-186

28. 28 http://sce.uhcl.edu/rungekutta feagin @ uhcl.edu Re(hλ) Im(hλ)