Extrapolation of explicit DIMSIMs of high order to solve the ordinary differential equations

Annie a/p Gorgey

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Type :	article
Subject :	Q Science (General)
Main Author :	Annie a/p Gorgey
Additional Authors :	Ali J. Kadhim
Title :	Extrapolation of explicit DIMSIMs of high order to solve the ordinary differential equations

Place of Production :	Tanjong Malim
Publisher :	Fakulti Sains dan Matematik
Year of Publication :	2019
Corporate Name :	Universiti Pendidikan Sultan Idris

Abstract : Universiti Pendidikan Sultan Idris

The purpose of this research is to investigate the e?ciency of ex-plicit diagonally implicit multi-stage integration methods with ex-trapolation. The author gave detailed explanation of explicit di-agonally implicit multi-stage integration method and compared thebase method with a technique known as extrapolation to improvethe e?ciency. Extrapolation for symmetric Runge-Kutta method isproven to improve the accuracy since with extrapolation the solu-tions exhibit asymptotic error expansion, however for General linearmethods, it is not known whether extrapolation can improve the ef-?ciency or not. Therefore this research focuses on the numericalexperimental results of the explicit diagonally implicit multistageintegration with and without extrapolation for solving some ordi-nary di?erential equations. The numerical results showed that thebase method with extrapolation is more e?cient than the methodwithout extrapolation

References

[1] Butcher, J.C. (1993) Diagonally-Implicit Multi-Stage IntegrationMethods. Applied Numerical Mathematics,11, 347-363.https://doi.org/10.1016/0168-9274(93)90059-Z

[2] Jackiewicz, J. (2009) General Linear Methods for Ordinary Dif-ferential Equations. John Wiley & Sons, Hoboken, NJ.https://doi.org/10.1002/9780470522165

[3] Butcher, J.C. (1987) The Numerical Analysis of Ordinary Dif-ferential Equations: Runge-Kutta and General Linear Methods.John Wiley & Sons, Chichester, New York.

[4] Butcher, J.C. and Jackiewicz J. (1998) Construction of HighOrder Diagonally Implicit Multistage Integration Methods forOrdinary-Di?erential Equations. Applied Numerical Mathemat-ics,27, 1-12. https://doi.org/10.1016/S0168-9274(97)00109-8

[5] Butcher, J.C. and Jackiewicz, J. (1996) Construction of Diagonal-ly Implicit General Linear Methods of Type 1 and 2 for OrdinaryDi?erential Equations. Applied Numerical Mathematics,21, 385-415. https://doi.org/10.1016/S0168-9274(96)00043-8

[6] Butcher, J.C., Chartier, P. and Jackiewicz, J. (1997) NordsieckRepresentation of DIMSIMs. Numerical Algorithms,16, 209-230.https://doi.org/10.1023/A:1019195215402

[7] Jackiewicz, Z. and Tracogna, S. (1995) A General Class of Two-Step Runge-Kutta Methods for Ordinary Di?erential Equations.SIAM Journal on Numerical Analysis,32, 1390-1427.https://doi.org/10.1137/0732064

[8] Cholloma, J. and Jackiewicz, Z. (2003) Construction of Two-StepRunge-Kutta Methods with Large Regions of Absolute Stability.Journal of Computational and Applied Mathematics,157, 125-137. https://doi.org/10.1016/S0377-0427(03)00382-0

[9] Conte, D., D’Ambrosio, R. and Jackiewicz, Z. (2010) Two-StepRunge-Kutta Methods with Quadratic Stability Functions. Jour-nal of Scienti?c Computing,44, 191-218.https://doi.org/10.1007/s10915-010-9378-x

[10] Wright, W. (2002) General Linear Methods with Inherent Runge-Kutta Stability. Ph.D. Thesis, The University of Auckland, Auck-land, New Zealand

[11] Jackiewicz, Z. (2005) Construction and Implementation of Gener-al Linear Methods for Ordinary Di?erential Equations: A Review.Journal of Scienti?c Computing,25, 29-49.https://doi.org/10.1007/s10915-004-4631-9

[12] Hairer, E. and Wanner, G. (1996) Solving Ordinary Di?erentialEquations II. Sti? and Di?erential-Algebraic Problems. Springer,Berlin. https://doi.org/10.1007/978-3-642-05221-7

[13] Butcher, J.C. (2006) General Linear Methods. Acta Numerica,15, 157-256. https://doi.org/10.1017/S0962492906220014

[14] Izzo, G. and Jackiewicz, Z. (2014) Construction of AlgebraicallyStable DIMSIMs. Journal of Computational and Applied Mathe-matics,261, 72-84. https://doi.org/10.1016/j.cam.2013.10.037

[15] Jackiewicz, Z. and Mittelmann, H. (2017) Construction of IMEXDIMSIMs of High Order and Stage Order. Applied NumericalMathematics,121, 234-248.https://doi.org/10.1016/j.apnum.2017.07.004

[16] Famelis, I.T. and Jackiewicz, Z. (2017) A New Approach to theConstruction of DIMSIMs of High Order and Stage Order. Ap-plied Numerical Mathematics,119, 79-93.https://doi.org/10.1016/j.apnum.2017.03.015

[17] Gorgey, A. (2012) Extrapolation of Symmetrized Runge-KuttaMethods. Ph.D. Thesis, The University of Auckland, NewZealand.

[18] Mona, T., Lagzi, I. and Havasi, A. (2015) Solving Reaction-Di?usion and Advection Problems with Richardson Extrapola-tion. Journal of Chemistry,2015, Article ID: 350362.https://doi.org/10.1155/2015/350362

[19] Mart´?n-Vaquero, J. and Kleefeld, B. (2016) Extrapolated Stabi-lized Explicit Runge-Kutta Methods. Journal of ComputationalPhysics,326, 141-155. https://doi.org/10.1016/j.jcp.2016.08.042

[20] Farago, I., Havasi, A. and Zlatev, Z. (2010) E?cient Implementa-tion of Stable Richardson Extrapolation Algorithms. Computersand Mathematics with Applications,60, 2309-2325.https://doi.org/10.1016/j.camwa.2010.08.025

[21] Butcher, J.C., Chartier, P. and Jackiewicz, Z. (1999) Experimentswith a Variable-Order Type 1 DIMSIM Code. Numerical Algo-rithms,22, 237-261. https://doi.org/10.1023/A:1019135630307DOI: 10.4236/jamp.2019.712212 3

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