EXTENDED MONO-IMPLICIT RUNGE-KUTTA METHODS FOR STIFF ODES
Keywords:
A-stability, stiff IVPs, order condition, Second derivative Mono-Implicit Runge-Kutta methodAbstract
An extended Mono Implicit Runge-kutta (EMIRK) method is considered herein for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equation (ODEs). The methods are A-stable for ???? = ????, ???? and ???????? . The ???? and ???? are the order of the input and output methods respectively. Numerical results are given to illustrate the application of the new methods.
Downloads
References
Cash J.R and Singhal A. Mono-Implicit Runge-Kutta formulae for numerical integration of stiff differential systems. IMA.J. Numer Anal, 2 (1982), 211-227.
J. C. Butcher,ImplicitRunge-Kuttaprocesses, math. Comp, 18 (1964).
Alexander. R, Diagonally implicit Runge–Kutta methods for stiff O.D. E’s. SIAM.J. Numer. Anal.vol 14, 6 (1977), 1006- 1021.
Norsett, S.P. semi-explicit runge-kutta methods, math.comput. no.6/74, university of Trondheim, 1974.
Jackiewicz Z, Renaut R.A, Zennaro M, Explicit two-step Runge-Kutta methods. Applications of Mathematics 40(6), (1995), pp. 433-456.
Burrage . K, Chipmanand. F.H and Muir. P.H., order results for Mono-Implicit Runge-Kutta methods SIAM J. Numer. Anal.31 (1994), 867-891.
De Meyer H, et al, On the generation of mono-implicit Runge-Kutta-Nystrom methods by mono-implicit Runge-Kutta methods. Journal of Computational and Applied Mathematics 111 (1999) 37–47.
Muir P, and Adams M, Mono-Implicit Runge-Kutta-Nystrom methods with Application to boundary value ordinary differential equations. BIT vol 41 4 (2001), 776-799.
Dow. F., Generalized Mono-Implicit Runge-Kutta Methods for Stiff Ordinary Differential Equations. Saint Marys University, Halifax, Nova Scotia, MSc Thesis (2017).
Enright, W.H., Second derivative multistep methods for stiff ODEs. SIAM.J.Numer. Anal. (1974) 11,321-331.
Chan R.P.K and Tsai A.Y.J. Explicit two-derivative Runge-Kutta methods. J. Numerical Algorithms, 53 (2010), 171-194.
Okuonghae, R.I {Variable order explicit second derivative general linear methods}. Comp. Applied Maths, Vol. 33, No. 1, (2014), pp. 243–255. See link.springer.com.
Turaci, M.O and Ozis, T. On explicit two-derivative two-step Runge-Kutta methods. Journal of Computational and Applied Mathematics (2018) See link.springer.com.
Aiguobasimwin,I.B, and Okuonghae, R.I. A Class of Two-Derivative Two-Step Runge-Kutta methods for Non-stiff ODEs. Hindawi. Journal of Applied Mathematics, (2019).
Butcher, J.C. and Hojjati, G. Second derivative methods with RK stability, Numer. Algor., 40 (2005) 415–429.
Abdi, A and Hojjati, G. An extension of general linear methods. Numer. Algor., 57(2011), pp.149–167.
Abdi, A and Hojjati, G. Higher order second derivative methods with Runge-Kutta stability for the numerical solution of stiff ODEs. Iranian J.Numer. Analysis and optimization, vol. 5, No 2 (2015), pp.1-10.
Okuonghae R.I. and M.N.OIkhile, M.N.O, L(α)-Stable Variable Order Second derivative Runge-Kutta methods . Numerical Analysis and Applications. Vol. 7, No 4, (2014), pp.314-327.
Okuonghae R.I. and M.N.OIkhile, M.N.O, Second derivative general linear methods. Numerical Algorithms. Vol. 67, issue 3, (2014), pp.637-654. Link.springer.com.
Ogunfeyitimi S.E. and Ikhile M. N. O, Generalized Second derivative linear multistep methods based on the methods of Enright. Int. J. Appl. comput.Maths vol. 25, No.4, 224- (2020).
Nwachukwu G.C. and Okor T, Second Derivative Generalized Backward Differentiation Formulae for Solving Stiff Problems. IAENG International Journal of Applied Mathematics, 48(1), (2018), 1-15.
Cash J.R. Second derivative extended backward differentiation formulas for the numerical integration of stiff systems}. SIAM J. Numer. Anal. 18(2) (1981) 21–36.
Okuonghae, R.I and Ikhile, M.N.O. Stable Multi-derivative GLM. Journal of Algorithms and Computational Technology Vol. 9 No. 4(2015), pp. 339-376.
Amodio. P and Mazzia.F, Boundary value methods based on Adams-type methods, Applied Numerical Mathematics 18(1995) 23-25.
Jator.S. and Sahi R., Bou-step Rndary value technique for initial value problems based on Adams-type second derivative methods, Int. J.Math.Educ.Sci. Educ.ifirst (2010), 1-8.
Downloads
Published
Issue
Section
License
Copyright (c) 2023 The Journals of the Nigerian Association of Mathematical Physics
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
