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

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

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4 h - the stepsize t0t0 t 0 + h where

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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

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6 for

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7 wherefor

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

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9 for all other values ofin the range for

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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.

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11 quadrature pointsQ for 0 12 13 14 15 16 24

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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

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13 11 8 3 5 non-matching pointsN for

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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

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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

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

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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)

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

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

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22 Variable step RK12(10) Pleiades problem GBS

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24 Kepler Problem (e = 0.1)

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26 Kepler Problem (e = 0.9)

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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

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

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