A-stable Two Derivative Mono-Implicit Runge-Kutta Methods for ODEs
Abstract
An A-stable Two Derivative Mono Implicit Runge-Kutta (ATDMIRK) method is considered herein for the numerical solution of initial value problems (IVPs) in ordinary differential equation (ODEs). The methods are of high-order A-stable for $p=q=\lbrace 2s+1\rbrace _{s=2}^{7}\ $ The $p$, $q$ and $s$ are the order of the input, output and the stages of the methods respectively. The numerical results affirm the superior accuracy of the newly develop methods compare to the existing ones.References
Abdi, A., & Hojjati, G. (2011). Maximal order for second derivative general linear methods with Runge-Kutta stability. Applied Numerical Mathematics, 61, 1046-1058. https://doi.org/10.1016/j.apnum.2011.06.004
Abdi, A., & Hojjati, G. (2011). An extension of general linear methods. Numerical Algorithms, 57, 149-167. https://doi.org/10.1007/s11075-010-9420-y
Abdi, A., & Hojjati, G. (2015). Higher order second derivative methods with Runge-Kutta stability for the numerical solution of stiff ODES. Iranian Journal of Numerical Analysis and Optimization, 5(2), 1-10.
Aiguobasimwin, I. B., & Okuonghae, R. I. (2019). A class of two-derivative two-step Runge-Kutta methods for non-stiff ODEs. Journal of Applied Mathematics, 2019, Article ID 2459809. https://doi.org/10.1155/2019/2459809
Aihie, I. B., & Okuonghae, R. I. (2022). Extended Mono-Implicit Runge-Kutta methods for stiff ODEs. Journal of the Nigerian Association of Mathematics and Physics (J.NAMP), 64, 53-58.
Alexander, R. (1977). Diagonally implicit Runge-Kutta methods for stiff O.D.E's. SIAM Journal on Numerical Analysis, 14 (6), 1006-1021. https://doi.org/10.1137/0714068
Amodio, P., & Mazzia, F. (1995). Boundary value methods based on Adams-type method. Appl. Numer. Math. 18, 23-25. https://doi.org/10.1016/0168-9274(95)00041-R
Bokhoven, W. M. G. (1980). Implicit end-point quadrature formulae. BIT, 3, 87-89.
Burrage, K., Chipman, F. H., & Muir, P. H. (1994). Order results for Mono-Implicit Runge-Kutta methods. SIAM Journal on Numerical Analysis, 31, 867-891. https://doi.org/10.1137/0731047
Butcher, J. C. (1964). Implicit Runge-Kutta processes. Mathematics of Computation, 18, 50-64. https://doi.org/10.2307/2003405
Butcher, J. C., Chartier, P., & Jackiewicz, Z. (1997). Nordsieck representation of DIMSIMs. Numerical Algorithms, 16, 209-230. https://doi.org/10.1023/A:1019195215402
Butcher, J. C., & Jackiewicz, Z. (1997). Implementation of diagonally implicit multistage integration methods for ordinary differential equations. SIAM Journal on Numerical Analysis, 34, 2119-2141. https://doi.org/10.1137/S0036142995282509
Butcher, J. C., & Hojjati, G. (2005). Second derivative methods with RK stability. Numerical Algorithms, 40, 415-429. https://doi.org/10.1007/s11075-005-0413-1
Cash, J. R. (1975). A class of Implicit Runge-Kutta methods for numerical integration of stiff differential systems. Journal of the ACM, 22, 504-511. https://doi.org/10.1145/321906.321915
Cash, J. R. (1981). Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM Journal on Numerical Analysis, 18(2), 21-36. https://doi.org/10.1137/0718003
Cash, J. R., & Singhal, A. (1982). Mono-Implicit Runge-Kutta formulae for numerical integration of stiff differential systems. IMA Journal of Numerical Analysis, 2, 211-227. https://doi.org/10.1093/imanum/2.2.211
Cash, J. R. (2003). Review paper. Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 459, 797-815. https://doi.org/10.1098/rspa.2003.1130
Chan, R. P. K., & Tsai, A. Y. J. (2010). Explicit two-derivative Runge-Kutta methods. Journal of Numerical Algorithms, 53, 171-194. https://doi.org/10.1007/s11075-009-9349-1
De Meyer, H., et al. (1999). On the generation of mono-implicit Runge-Kutta-Nystrom methods by mono-implicit Runge-Kutta methods. Journal of Computational and Applied Mathematics, 111, 37-47. https://doi.org/10.1016/S0377-0427(99)00130-2
Dow, F. (2017). Generalized Mono-Implicit Runge-Kutta Methods for Stiff Ordinary Differential Equations (MSc Thesis). Saint Mary's University, Halifax, Nova Scotia.
Ehigie, J. O., Jator, S. N., Sofoluwe, A. B., & Okunuga, S. A. (2014). Boundary value technique for initial value problems with continuous second derivative multistep method of Enright. Computers & Mathematics with Applications, 33(1), 81-93. https://doi.org/10.1007/s40314-013-0044-4
Enright, W. H. (1974). Second derivative multistep methods for stiff ODEs. SIAM Journal on Numerical Analysis, 11, 321-331. https://doi.org/10.1137/0711029
Hairer, E., & Wanner, G. (1996). Solving ordinary differential equations II: Stiff and differential algebraic problems (2nd rev. ed.). Springer Verlag.
Jator, S., & Sahi, R. (2010). Boundary value technique for initial value problems based on Adams-type second derivative methods. International Journal of Mathematical Education in Science and Technology, 41 (6), 819-826.
Muir, P., & Adams, M. (2001). Mono-Implicit Runge-Kutta-Nystrom methods with Application to boundary value ordinary differential equations. BIT, 41(4), 776-799. https://doi.org/10.1023/A:1021956304751
Muir, P., & Owen, B. (1993). Order Barriers and Characterizations for Continuous Mono-Implicit Runge-Kutta schemes. Mathematics of Computation, 61 (204), 675-699. https://doi.org/10.1090/S0025-5718-1993-1195425-8
Jackiewicz, Z., Renaut, R. A., & Zennaro, M. (1995). Explicit two-step Runge-Kutta methods. Applications of Mathematics, 40(6), 433-456. https://doi.org/10.21136/AM.1995.134306
Norsett, S. P. (1974). Semi-explicit Runge-Kutta methods. Mathematics and Computation, No.6/74. University of Trondheim.
Nwachukwu, G. C., & Okor, T. (2018). Second Derivative Generalized Backward Differentiation Formulae for Solving Stiff Problems. IAENG International Journal of Applied Mathematics, 48(1), 1-15.
Ogunfeyitimi, S. E., & Ikhile, M. N. O. (2020). Generalized second derivative linear multistep methods based on the methods of Enright. International Journal of Applied and Computational Mathematics, 6(76). https://doi.org/10.1007/s40819-020-00827-0
Ogunfeyitimi, S. E., & Ikhile, M. N. O. (2021). Implicit-explicit second derivative LMM for stiff ordinary differential equations. Journal of the Korean Society for Industrial and Applied Mathematics, 25(4), 224-261.
Okuonghae, R. I., & Aiguobasimwin, I. B. (2018). High-order hybrid Obreshkov multistep methods. IAENG Journal of Applied Mathematics, 48(1), 73-83.
Okuonghae, R. I., & Ikhile, M. N. O. (2014). L(a)-Stable variable second-derivative Runge-Kutta methods. Numerical Analysis and Applications, 7(4), 314-327 https://doi.org/10.1134/S1995423914040065
Okuonghae, R. I., & Ikhile, M. N. O. (2014). Second derivative general linear methods. Numerical Algorithms, 67(3), 637-654. https://doi.org/10.1007/s11075-013-9814-8
Okuonghae, R. I., & Ikhile, M. N. O. (2015). L(a)-stable multi-derivative GLM. Journal of Algorithms and Computational Technology, 9(4), 339-376. https://doi.org/10.1260/1748-3018.9.4.339
Okuonghae, R. I. (2014). Variable order explicit second derivative general linear methods. Computational and Applied Mathematics, 33(1), 243-255. https://doi.org/10.1007/s40314-013-0058-y
Abu Arqub, O. (2018). Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. International Journal of Numerical Methods for Heat & Fluid Flow, 28, 828-856. https://doi.org/10.1108/HFF-07-2016-0278
Abu Arqub, O., Tayebi, S., Baleanu, D., Baleanu, M. S., Mahmoud, W., & Alsulami, H. (2022). A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms. Results in Physics, 41, 105912. https://doi.org/10.1016/j.rinp.2022.105912
Abu Arqub, O. (2020). Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method. International Journal of Numerical Methods for Heat & Fluid Flow, 30(11), 4711-4733. https://doi.org/10.1108/HFF-10-2017-0394
Otunta, F. O., Ikhile, M. N., & Okuonghae, R. I. (2007). Second derivative continuous linear multistep methods for the numerical integration of stiff system of ordinary differential equations. Journal of the Nigerian Association of Mathematical Physics, 11, 159-174. https://doi.org/10.4314/jonamp.v11i1.40210
Turaci, M. O., & Ozis, T. (2018). On explicit two-derivative two-step Runge-Kutta methods. Comp. Appl. Math., 37, 6920-6954. https://doi.org/10.1007/s40314-018-0719-y
Akinfenwa, O. A., & Jator, S. N. (2015). Extended continuous block backward differentiation formula for stiff systems. Fasciculi Mathematici, 55, 5-18. https://doi.org/10.1515/fascmath-2015-0010
Okor, T., & Nwachukwu, G. C. (2022). High order extended boundary value methods for the solution of stiff systems of ODEs. Journal of Computational Applied Mathematics, 400, 113750. https://doi.org/10.1016/j.cam.2021.113750
Sweis, H., Shawagfeh, N., & Abu Arqub, O. (2022). Fractional crossover delay differential equations of Mittag-Leffler kernel: Existence, uniqueness, and numerical solutions using the Galerkin algorithm based on shifted Legendre polynomials. Results in Physics, 41, 105891. https://doi.org/10.1016/j.rinp.2022.105891
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