One-step 9-stage Hermite–Birkhoff–Taylor ODE solver of order 10
Tóm tắt
A one-step 9-stage Hermite–Birkhoff–Taylor method of order 10, denoted by HBT(10)9, is constructed for solving nonstiff systems of first-order differential equations of the form y′=f(x,y), y(x
0)=y
0. The method uses y′ and higher derivatives y
(2) to y
(4) as in Taylor methods and is combined with a 9-stage Runge–Kutta method. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to Taylor- and Runge–Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than Dormand–Prince DP(8,7)13M. The stepsize is controlled by means of y
(2) and y
(4). HBT(10)9 is superior to DP(8,7)13M and Taylor method of order 10 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. These numerical results show the benefits of adding high-order derivatives to Runge–Kutta methods.
Tài liệu tham khảo
Barrio, R.: Sensitivity analysis of ODEs/DAEs using the Taylor series method. SIAM J. Sci. Comput. 27(6), 1929–1947 (2006)
Barrio, R., Blesa, F., Lara, M.: VSVO formulation of the Taylor method for the numerical solution of ODEs. Comput. Math. Appl. 50, 93–111 (2005)
Berntsen, J., Espelid, T.O.: Error estimation in automatic quadrature routines. ACM Trans. Math. Softw. 17, 233–255 (1991)
Butcher, J.C.: On Runge–Kutta processes of high order. J. Aust. Math. Soc. 18, 50–64 (1964)
Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Wiley, London (1987)
Corliss, G.F., Chang, Y.F.: Solving ordinary differential equations using Taylor series. ACM Trans. Math. Softw. 8(2), 114–144 (1982)
Davis, P.J., Rabinowitz, P.: Numerical Integration. Blaisdell, Waltham (1967)
Deprit, A., Zahar, R.M.W.: Numerical integration of an orbit and its concomitant variations. Z. Angew. Math. Phys. 17, 425–430 (1966)
Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer, Berlin (1993). Section III.8
Hajji, M.A., Vaillancourt, R.: Matrix derivation of Gaussian quadratures. Sci. Proc. Riga Techn. Univ. 29(48), 198–213 (2006)
Hoefkens, J., Berz, M., Makino, K.: Computing validated solutions of implicit differential equations. Adv. Comput. Math. 19, 231–253 (2003)
Lara, M., Elipe, A., Palacios, M.: Automatic programming of recurrent power series. Math. Comput. Simul. 49, 351–362 (1999)
Nedialkov, N.S., Jackson, K.R., Corliss, G.F.: Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105, 21–68 (1999)
Piessens, R., de Doncker-Kapenga, E., Überhuber, C.W., Kahaner, D.K.: QUADPACK. A Subroutine Package for Automatic Integration. Springer Series in Comput. Math., vol. 1. Springer, Berlin (1983)
Prince, P.J., Dormand, J.R.: High order embedded Runge–Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981)
Rabe, E.: Determination and survey of periodic Trojan orbits in the restricted problem of three bodies. Astron. J. 66(9), 500–513 (1961)
Sharp, P.W.: Numerical comparison of explicit Runge–Kutta pairs of orders four through eight. ACM Trans. Math. Softw. 17, 387–409 (1991)
Steffensen, J.F.: On the restricted problem of three bodies. Danske Vid. Selsk., Mat.-fys. Medd. 30(18), 17 (1956)