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

High-Order Explicit Runge-Kutta Methods Using m -Symmetry. T. Feagin University of Houston – Clear Lake Houston, Texas, USA June 24, 2009. Background and introduction The Runge-Kutta equations of condition New variables Reformulated equations m -symmetry

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

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

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

  3. The Initial Value Problem

  4. Explicit Runge-Kutta Methods t0 t0+ h where h - the stepsize

  5. Explicit Runge-Kutta Methods

  6. Equations of Condition for

  7. New Variables for for for where for for for

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

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

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

  11. m-symmetry Q quadrature points 0 12 13 14 15 16 24 for for

  12. 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

  13. m-symmetry N non-matching points 11 8 3 5 for

  14. New Equations for a 6th Order Method

  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 αk2 (the first internal quadrature point) such that the order at the quadrature points is m/2 Finding an m-symmetric Method

  16. Identify the matching and non-matching points Obtain values for any of the αk‘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 Finding an m-symmetric Method (cont’d)

  17. Plots Showing m-symmetry rk4 vsk pk,6,21vsk Example plots for the 12th-order method

  18. Seeking to reduce the local truncation errors by minimizing size and number of the unsatisfied 13th-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 RK12(10) - Optimizing the Method Im(hλ) Re(hλ)

  19. Efficiency Diagram for Kepler Problem -log10(error) RK12 Fixed step integration RK10H Eccentricity = 0.4 RK8CV RK6B RK4 log10(NF)

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

  21. The Pleiades Problem

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

  23. Comparisons of High-Order Methods

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

  25. Comparisons of High-Order Methods

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

  27. 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

  28. Where to Obtain Coefficients, etc. http://sce.uhcl.edu/rungekutta feagin@uhcl.edu Im(hλ) Re(hλ) Re(hλ)

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