AN EFFICIENT FAMILY OF SECOND DERIVATIVE RUNGE-KUTTA COLLOCATION METHODS FOR OSCILLATORY SYSTEMS
Abstract
An efficient family of high-order second derivativeRunge-Kutta collocation methods is derived for the numerical
solution of oscillatory systems.
The approach uses polynomial interpolation and collocation techniques to construct continuous schemes which were evaluated at both step and off-step points to obtain hybrid formulae.
The hybrid formulae can be applied simultaneously as block methods for moving the integration process forward at a time, if desired.
The block methods based on hybrid formulation can also be converted to second derivative Runge-Kutta collocation methods.
The stability properties and order of accuracy of the methods are
studied. They can also be implemented easily since they are
collocation methods and provide a high order of accuracy. The
methods were illustrated by the applications to some test problems
of oscillatory system found in the literature and the numerical
results obtained confirm the accuracy and efficiency of the methods.
References
bibitem[1]{journ2.1}K.Burrage, J.C.Butcher, {it Nonlinear stability for a general class of differential equation
methods}, BIT Numer. Math. textbf{20} 185 -203, 1979.
bibitem[2]{journ2.2}S.O. Fatunla, {it Numerical integrators for stiff and highly oscillatory
differential equations}, J. Math. Comp textbf{34} (150) 373-390,
bibitem[3]{journ2.3}L.R. Petzold, L.O. Jay, J.Yen, {it Numerical solution of highly oscillatory ordinary differential
equations}, Acta Numerica. textbf{6} 437-483, 1997.
bibitem[4]{journ2.4}F. Costabile, A. Napoli, {it Stability of Chebychev collocation methods}, Comput. Math. Appl.
textbf{47} , 659-666, 2004.
bibitem[5]{journ2.5}F. Costabile, A. Napoli, {it A class of collocation methods for numerical integration of
initial value problems}, Comput. Math. Appl. textbf{62} 3221-3235,
bibitem[6]{journ2.6}M. Urabe, {it An implicit one-step method of high-order accuracy for the
numerical integration of ordinary differential equations}, J. Numer.
Math. textbf{15} 151 - 164, 1970.
bibitem[7]{journ2.7}H. Shantani, {it On one-step methods utilizing the second derivative}, Hiroshima Math
textbf{1} 349 - 372, 1971.
bibitem[8]{journ2.8}H. Shantani,, {it On explicit one-step methods utilizing the second
derivative}, Hiroshima Math. textbf{2} 353 - 368, 1972.
bibitem[9]{journ2.9}J.R.Cash, {it High order methods for numerical integration of ordinary
differential equations}, Numer. Math. textbf{30} 385 - 409, 1978.
bibitem[10]{journ2.10}T. Mitsui, {it Runge-Kutta type integration formulas including the evaluation of
the second derivative Part I}, Publ. RIMS Kyoto University ,Japan
textbf{18} 325 - 364, 1982.
bibitem[11]{journ2.11}S.D. Capper, J.R. Cash, D. R. Moore, {it Lobatto-Obrechkoff formulae for 2nd , order two-points boundary
value problems}, J. Numer. Anal. Indus. and Appl. Math. JNAIAM
textbf{314} 13 - 25, 2006.
bibitem[12]{journ2.12}R.P.K. Chan, A.Y.J. Tsai, {it On explicit two-derivative Runge-Kutta methods}, J. Numer. Algor.
textbf{53} 171 - 194, 2010.
bibitem[13]{journ2.13}D.G. Yakubu, S. Markus, {it Second derivative of high order accuracy methods for the numerical
integration of stiff initial value problems}, Afrika Math.
textbf{27} (5-61) 963-977, 2016.
bibitem[14]{journ2.14}J.C. Butcher, {it Numerical Methods for Ordinary Differential Equations}, 2nd edn. Wiley Chichester, 2008.
bibitem[15]{journ2.15}T.Mitsui, D.G. Yakubu, {it Two-step family of look-ahead linear multistep methods for ODEs}, The science and engineering review Doshisha University Japan
textbf{52} (3) 181-188, 2011.
bibitem[16]{journ2.16}P. Onumanyi, D.O. Awoyemi, S.N. Jator, U.W. Sirisena {it New linear multistep methods with continuous coefficients for
first order initial value problems}, J. Nig. Math. Soci. textbf{13}
-51, 1994.
bibitem[17]{journ2.17}I. Lie, S. P. N{o}rsett, {it Superconvergence for multistep
collocation}, Math. Comp. textbf{52} 65-80, 1989.
bibitem[18]{journ2.18}D.G. Yakubu, M. Aminu, A. Aminu, {it The numerical integration of stiff systems using stable
multi-step multi-derivative methods}, J. Mod. Meth. Numer. Math.,
textbf{8} (1-2) 99-117, 2017.
bibitem[19]{journ2.19}J. Vigo-Aguiar, H. Ramos, {it A family of A-stable Runge-Kutta collocation methods of high
order for initial value problems}, IMA J. Numer. Anal. textbf{27}
(4) 798-817, 2007.
bibitem[20]{journ2.20}J. D. Lambert, {it Computational Methods in Ordinary Differential Equations}, John Wiley and Sons Ltd, 1973.
bibitem[21]{journ2.21}G. Dahlquist, {it 'A special stability problem for linear multistep
methods'}, BIT J. Numer. Math. textbf{3} 27-43, 1963.
bibitem[22]{journ2.22}H. Ramos, V. Jesus, {it A fourth-order Runge-Kutta method based on
BDF-type Chebychev approximations}, J. Comp. Appl. Math.
textbf{204} 124-136, 2007.
bibitem[23]{journ2.23}J. D. Lambert, {it Numerical Methods for Ordinary Differential Systems}, John Wiley and Sons, Ltd, 1991.
bibitem[24]{journ2.24}W. C. Gear, {it DIFSUB for solution of ordinary differential equations}, Comm ACM.
textbf{14} 185-190, 1971.
bibitem[25]{journ2.25}E. Sch$ddot{a}$fer, Br. Freiburi, {it A new approach to explain the
"High Irradiance Responses" of photomorphoenesis on the basis of
phytochrome}, J. Math. Bio. textbf{2} 41-56, 1975.
