|
UPSI Digital Repository (UDRep)
|
|
| ||||||||||||||||||||||||
| Abstract : Universiti Pendidikan Sultan Idris |
| Diagonally implicit multistage integration methods (DIMSIMs) are widely utilized in finding the solution to any problems in the subject of ordinary differential equations. These methods are selected from the general linear methods, which is considerable potential for efficient implementations. The extrapolation is derived from the stability of the explicit Runge-Kutta methods. In this paper, the combination of DIMSIMs with Richardson extrapolation of different orders shows that numerical solutions give higher accuracy when the extrapolation is applied with the base method. 2023 Penerbit UTM Press. |
| References |
Abdi, A. and Jackiewicz, Z. 2019. Towards a Code for Nonstiff Differential Systems based on General Linear Methods with Inherent Runge-Kutta Stability. Applied Numerical Mathematics. 136: 103-121. Doi: https://doi.org/10.1016/j.apnum.2018.10.001. Butcher, J. and Jackiewicz, Z. 1993. Diagonally Implicit General Linear Methodsfor Ordinary Differential Equations. BIT. 33(3): 452-472. DOI: https://doi.org/10.1007/BF01990528. Butcher, J. C. 1993. Diagonally-implicit Multi-stage Integration Methods. APP. Numer. Alg. 11: 347-363. Doi: https://doi.org/10.1016/0168-9274(93)90059-Z. Butcher, J. C. 2016. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. John Wiley and Sons, New York. DOI: https://doi.org/10.1007/978-3-319-17689-5_2. Butcher, J. C., Chartier, P. and Jackiewicz, Z. 1997. Nordsieck Representation of DIMSIMs. Numerical Algorithms.16: 209-230. Doi: https://doi.org/10.1023/A:1019195215402. Butcher, J. C., Chartier, P. and Jackiewicz, Z. 1999. Experiments with a Variable-order Type 1 DIMSIMs Code. Numerical Algorithms. 22: 273-261. Doi: https://doi.org/10.1023/A:1019135630307. Butcher, J. C. and Jackiewicz, Z. 1996. Construction of Diagonally Implicit General Linear Methods of Type 1 and 2 for Ordinary Differential Equations. Appl. Numer. Math. 21: 385-415. Doi: http://doi.org/10.1016/S0168-9274(96)00043-8. Califano, G., Izzo, G. and Jackiewicz, Z. 2017. Starting Procedures for General Linear Methods. Applied Numerical Mathematics. 120: 165-175. Doi: https://doi.org/10.1016/j.apnum.2017.05.009. Dahlquist, G. 1963. A Special Stability for Linear Multistep Methods. BIT. 3: 27-43. Doi: http://doi.org/10.1007/BF01963532. 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. DOI: http://doi.org/10.1016/j.apnum.2017.03.015. Huang, S. J. 2005. Implementation of General Linear Methods for Stiff Ordinary Differential Equation. Ph.D. Thesis. University of Auckland, New Zealand. Hull, T. E., Enright , W. H., Fellen, B. M. and Sedgwick, A. E. 1972. Comparing Numerical Methods for Ordinary Differential Equations. SIAM Journal on Numerical Analysis. 9(4): 603-637. DOI: http://doi.org/10.1137/0709052. Mahdi, H., Hojjati, G., & Abdi, A. 2019. Explicit General Linear Methods with a Large Stability Region for Volterra Integro-differential Equations. Mathematical Modelling and Analysis. 24(4): 478-493. DOI: https://doi.org/10.3846/mma.2019.029. Ramazani, P., Abdi, A., Hojjati, G., & Moradi, A. 2022. Explicit Nordsieck Second Derivative General Linear Methods For ODEs. The ANZIAM Journal. 64(1): 69-88. Doi: https://doi.org/10.1017/S1446181122000049. Richardson, L. F. 1911. The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equation, with an Application to the Stresses in Amasonry Dam. Philos. Trans. Roy. Soc. London, ser. A. 210: 307-857. Wright, W. 2002. General Linear Methods with Inherent Runge-Kutta Stability. Ph.D. thesis, University of Auckland, New Zealand. |
| This material may be protected under Copyright Act which governs the making of photocopies or reproductions of copyrighted materials. You may use the digitized material for private study, scholarship, or research. |