High-Order Explicit Runge-Kutta Methods Using

m

-Symmetry T. Feagin University of Houston – Clear Lake Houston, Texas, USA June 24, 2009 1

Overview • • • • • • • Background and introduction The Runge-Kutta equations of condition New variables Reformulated equations

m

-symmetry Finding an

m

-symmetric method Numerical experiments 2

The Initial Value Problem 3

Explicit Runge-Kutta Methods t 0 h - the stepsize t 0 + h

where

4

Explicit Runge-Kutta Methods The order of the formula m The number of new equations of order m 6 7 8 9 10 11 12 13 14 1 2 3 4 5 20 48 115 286 719 1842 4766 12486 32973 1 1 2 4 9 15 87811 The total number of equations for order m 1 2 4 8 17 37 85 200 486 1205 3047 7813 20299 53272 141083 Number of stages n The number of unknowns n (n+1)/2 7 9 11 15 17 25 35 1 2 3 4 6 28 45 66 120 153 325 630 1 3 6 10 21 5

Equations of Condition for 6

New Variables for for for where for for for 7

New Variables for one of the row simplifying assumptions when zero for where for for for one of the column simplifying assumptions when zero for 8

Reformulated Equations of Condition for for all other values of in the range 9

m

-symmetry Theorem: Any

m

-symmetric Runge-Kutta method is of order

m

.

The set of integer subscripts can be partitioned into three subsets

Q quadrature

points

M matching

points

N non-matching

points 10

m

-symmetry 0 12 13 14 15 16 24

Q quadrature

points for for 11

m

-symmetry

M matching

points 1 7 4 2 6 9 10 23 19 21 22 20 18 17 for for where and is the smallest value of such that 12

m

-symmetry

N non-matching

points 11 8 3 5 for 13

New Equations for a 6 th Order Method 14

Finding an

m

-symmetric Method • • • 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 15

Finding an

m

-symmetric Method (cont’d) • • • • 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 16

Plots Showing

m

-symmetry

r k4

vs

k p k,6,21

vs

k

Example plots for the 12 th -order method 17

RK12(10) - Optimizing the Method • • • 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 Im(hλ) Re(hλ) 18

35 30 25 20 15 10 5 Efficiency Diagram for Kepler Problem E Fixed step integration Eccentricity = 0.4

RK12 RK10H RK8CV RK6B RK4 5 log 10 NF log10(NF) 2.5

3 3.5

4 4.5

19

Estimation of Local Truncation Errors The true error and the estimated error for RK12(10) 20

The Pleiades Problem 21

Comparison with Extrapolation Method RK12(10) GBS Variable step Pleiades problem 22

Comparisons of High-Order Methods 23

Comparisons of High-Order Methods Kepler Problem (e = 0.1) 24

Comparisons of High-Order Methods 25

Comparisons of High-Order Methods Kepler Problem (e = 0.9) 26

References W. B. Gragg,

On extrapolation algorithms for ordinary initial value problems

. SIAM J. Num. Anal., 2 (1965) pp. 384-403 E. Baylis Shanks,

Solutions of Differential Equations by Evaluations of Functions

, Math. Comp. 20, No. 93 (1966), pp. 21-38 E. Fehlberg,

Classical Fifth-, Sixth-, Seventh- , and Eighth-Order Runge-Kutta Formulas with Stepsize Control

, NASA TR R-287, (1968) E. Hairer,

A Runge-Kutta method of order 10

, J. Inst. Math. Applics. 21 (1978) pp. 47-59 Hiroshi Ono,

On the 25 stage 12th order explicit Runge--Kutta method

, JSIAM Journal, Vol. 16, No. 3, 2006, p. 177-186 J.H. Verner,

The derivation of high order Runge Kutta methods,

Univ. of Auckland, New Zealand, Report No. 93, (1976) P.J. Prince and J.R. Dormand,

High-order embedded Runge-Kutta formulae

, J Comput. Appl. Math., 7 (1981), pp. 67-76 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 27

Where to Obtain Coefficients, etc.

http://sce.uhcl.edu/rungekutta feagin @ uhcl.edu

Im(hλ) 28