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 JAMP  Vol.7 No.12 , December 2019
Extrapolation of Explicit DIMSIMs of High Order to Solve the Ordinary Differential Equations
Abstract: The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit multi-stage integration method and compared the base method with a technique known as extrapolation to improve the effciency. Extrapolation for symmetric Runge-Kutta method is proven to improve the accuracy since with extrapolation the solutions exhibit asymptotic error expansion, however for General linear methods, it is not known whether extrapolation can improve the effciency or not. Therefore this research focuses on the numerical experimental results of the explicit diagonally implicit multistage integration with and without extrapolation for solving some ordinary differential equations. The numerical results showed that the base method with extrapolation is more effcient than the method without extrapolation.
Cite this paper: Kadhim, A. and Gorgey, A. (2019) Extrapolation of Explicit DIMSIMs of High Order to Solve the Ordinary Differential Equations. Journal of Applied Mathematics and Physics, 7, 3022-3030. doi: 10.4236/jamp.2019.712212.
References

[1]   Butcher, J.C. (1993) Diagonally-Implicit Multi-Stage Integration Methods. 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 Differential Equations. John Wiley & Sons, Hoboken, NJ.
https://doi.org/10.1002/9780470522165

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

[4]   Butcher, J.C. and Jackiewicz J. (1998) Construction of High Order Diagonally Implicit Multistage Integration Methods for Ordinary-Differential Equations. Applied Numerical Mathematics, 27, 1-12.
https://doi.org/10.1016/S0168-9274(97)00109-8

[5]   Butcher, J.C. and Jackiewicz, J. (1996) Construction of Diagonally Implicit General Linear Methods of Type 1 and 2 for Ordinary Differential 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) Nordsieck Representation 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 Differential 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-Step Runge-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-Step Runge-Kutta Methods with Quadratic Stability Functions. Journal of Scientific 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, Auckland, New Zealand.

[11]   Jackiewicz, Z. (2005) Construction and Implementation of General Linear Methods for Ordinary Differential Equations: A Review. Journal of Scientific Computing, 25, 29-49.
https://doi.org/10.1007/s10915-004-4631-9

[12]   Hairer, E. and Wanner, G. (1996) Solving Ordinary Differential Equations II. Stiff and Differential-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 Algebraically Stable DIMSIMs. Journal of Computational and Applied Mathematics, 261, 72-84.
https://doi.org/10.1016/j.cam.2013.10.037

[15]   Jackiewicz, Z. and Mittelmann, H. (2017) Construction of IMEX DIMSIMs of High Order and Stage Order. Applied Numerical Mathematics, 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 the Construction of DIMSIMs of High Order and Stage Order. Applied Numerical Mathematics, 119, 79-93.
https://doi.org/10.1016/j.apnum.2017.03.015

[17]   Gorgey, A. (2012) Extrapolation of Symmetrized Runge-Kutta Methods. Ph.D. Thesis, The University of Auckland, New Zealand.

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

[19]   Martín-Vaquero, J. and Kleefeld, B. (2016) Extrapolated Stabilized Explicit Runge-Kutta Methods. Journal of Computational Physics, 326, 141-155.
https://doi.org/10.1016/j.jcp.2016.08.042

[20]   Farago, I., Havasi, A. and Zlatev, Z. (2010) Efficient Implementation of Stable Richardson Extrapolation Algorithms. Computers and 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) Experiments with a Variable-Order Type 1 DIMSIM Code. Numerical Algorithms, 22, 237-261.
https://doi.org/10.1023/A:1019135630307

 
 
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