THIRD DERIVATIVE MONO-IMPLICIT RUNGE-KUTTA METHODS FOR STIFF ODEs
DOI:
https://doi.org/10.60787/tnamp.v23.623Keywords:
Third Derivative Mono-Implicit Runge-kutta, order condition, A-stability, stiff IVPsAbstract
This work focuses on a Third-Derivative Mono-Implicit Runge–Kutta (TD-MIRK) method developed for the numerical approximation of stiff initial value problems (IVPs) in ordinary differential equations. The order conditions for the scheme are derived using Taylor series expansions. We introduce a seventh-order TD-MIRK method constructed to require the least possible number of function evaluations. The numerical experiments are then compared with well-known methods previously reported in the literature.
Downloads
References
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).
Aihie, I.B, and Okuonghae, R.I. A-stable Two Derivative Mono-Implicit Runge-Kutta Methods for ODEs. Earthline Journal of Mathematical Sciences 14(3), (2024),565-588.
Aihie, I.A and Okuonghae, R.I. Extended Mono-Implicit Runge-Kutta methods for stiff ODEs. Journals of Nigerian Association of Mathematical Physics 64, (2022), pp.53-58.
Aihie, I.A and Okuonghae, R.I. Second-Derivative Two-Step Mono-Implicit Runge-Kutta methods for stiff [ODEs.International](http://ODEs.International) journals of Mathematics Trends and Technology IJMTT 70,2024. Physics 64, (2022), pp.53-58.
Adoghe, L.O, Omole, E.O and fadughba, S.E. Third derivative method for solving stiff system of ordinary differential equations. Int.J. mathematics in operational Research Vol.23, No.3(2022), pp.412-425.
Akinfenwa, O.A. Third derivative hybrid block integrator for solution of stiff system of initial value problems. African Mathematical Union and Springer-Verlag Berlin Heidel berg (2017) Vol.2, pp.629-641.
Burrage, K. Chipman F.H and Muir P.H. Muir order results for Mono-Implicit Runge Kutta methods. SIAM J. Numer. Anal.31 (1994), 867-891.
Butcher J.C. Implicit Runge-Kutta processes. Math. comp 18(1964).
Butcher, J.C, Chartier, P and Jackiewicz, Z. Nordsieck representation of DIM SIMs,Numer. Algor., 16 (1997) 209–230.
Butcher, J.C and Jackiewicz, Z. Implementation of diagonally implicit multistage inte gration methods for ordinary differential equations, SIAM J.Numer. Anal., 34 (1997) 2119–2141.
Cash J.R. A class of Implicit Runge-Kutta methods for numerical integration of stiff differential systems. J. ACM, 22 (1975), 504-511.
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.
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 Mathe matics 111 (1999) 37–47.
Dow F., Generalized Mono-Implicit Runge-Kutta Methods for Stiff Ordinary Differential Equations. Saint Marys University, Halifax, Nova Scotia, MSc Thesis (2017).
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.
Muir P, and Owen B, Order Barriers and Characterizations for Continuous Mono Implicit Runge-Kutta schemes. Math. Comp. vol.61 204(1993), 675-699.
Okuonghae, R.I and Ikhile, M.N.O. L(α)-Stable variable second-derivative Runge-Kutta methods Numerical Analysis and Applications. Vol. 7, No 4, (2014), pp. 314-327.
Okuonghae, R.I and Aiguobasimwin, I.B. Variable step-size Implementation of Hybrid Linear Multustep methods.Journals of Nigerian Association of Mathematical Physics 27,(2014),pp.29-36.
Okuonghae, R.I and Ikhile, M.N.O. L(α)-Stable Multi-derivative GLM. Journal of Al gorithms and Computational Technology Vol. 9 No. 4(2015), pp. 339-376
Downloads
Published
Issue
Section
License
Copyright (c) 2025 The Transactions of the Nigerian Association of Mathematical Physics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
