A-Stable Block ETRs/ETR2s Methods for Stiff ODEs

Yohanna Sani Awari

Abstract


We present a class of fourth and sixth order A-stable block extended trapezoidal rule of first kind (ETRs) and extended trapezoidal rule of second kind (ETR2s) methods which are found to be adequate for the numerical integration of stiff ordinary differential equations. The single continuous formulation of this methods are evaluated at some grid and interior points yielding the multi-discrete schemes which are implemented in block form thereby generating simultaneously approximate solutions  at once without recourse to predictors. By this approach, the need for starters is eliminated. The stability properties of the block ETRs/ETR2s methods discussed and were shown to preserve the A-stability property of the trapezoidal rule, their absolute stability regions also presented. The newly derived block methods were implemented on five stiff systems of ordinary differential equations occurring in real life to show efficiency and accuracy.


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References


Areo, E.A.and Adeniyi, R.B. Block Implicit One Step Method for the Numerical Integration of Initial Value Problems in Ordinary Differential Equations, International Journal of Mathematics and Statistics Studies, 2014, 2(3): 4-13.

Awoyemi, D.O., Ademiluyi, R.A. and Amuseghan, E. Off-grid exploitation in the development of more accurate method for the solution of ODEs, Journal of Mathematical Physics, 2007, 12: 379-386.

Butcher, J.C. Numerical Methods for Ordinary differential Equuations, 2008, John Wiley & sons, England

Brugnano, L. and Trigiante, D. (1998). Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon Breach Publishers, Amsterdam. 418p.

Dahlquist, G. (1956). Numerical Integration of Ordinary Differential Equations, Math. Scand., 4: 33-50.

Dahlquist, G. (1963). A Special Stability Problem for Linear Multistep Methods, BIT. 3: 27-43.

Enright, W.H. Second Derivative Multistep Methods for Stiff Ordinary Differential Equations, SIAM, J. Num. Anal., 1974, 11:321-331.

Fatunla, S.O. (1988). Numerical Methods for IVPs in Ordinary Differential Equations. Academic Press Inc. Harcourt Brace Jovanivich Publishers, New York.

Henrici,P. (1962). Discrete variable methods for ODE’s.John Wiley, New York.

Ishak, F. and Siti, N.A. (2016). Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations: World Academy of Science, Engineering and Technology; International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 10(11), 2016.

Jingjun, Zhao et al., (2014). Delay-dependent stability analysis analysis of symmetric boundary value methods for linear delay integro-differential equations. Springer Links, Numerical Algorithms, 65(1), 125-151.

Lambert, J.D. (1973). Computational Methods in Ordinary Differential Equations. New York, USA: John Wiley. 278p

Lambert, J.D. (1991). Numerical Methods for Ordinary Differential Systems of initial Value Problems, John Wiley, New York.

Mehdizadeh, K.M., Nasehi, O.N. and Hojjati, G. A Class of second derivative multistep methods for stiff systems. Acta Universitatis Apulensis, 2012, No. 30: 171-188.

Miletics, E. and Moln´arka, G. Taylor Series Method with Numerical Derivatives for Numerical Solution of ODE Initial Value Problems, HU ISSN 1418-7108: HEJ Manuscript no.: ANM-030110-B, 2009.

Murray, J.B. (1977). Lectures on nonlinear-differential-equation models in biology, Clarendon Press, Oxford.

Okunuga, S.A. and Ehigie, J. A new Derivation of Continuous Collocation Multistep Methods Using Power Series as Basis Function, Journal of Modern Mathematical and Statistics, 2009, 3(2): 43-50.

Rei-Wei Song and Ming-Gong Lee. A New Block Method and Their Application to Numerical Solution of Ordinary Differential Equations. D09424011@chu.edu.tw, mglee@chu.edu.tw.

Xiao-Hui Liu, Yu-Jiang Wu, Jin-Yun Yuan, Raimundo J. B.de Sampaio, Yan-TaoWang (2015). Sixth-order Compact Extendent Trapezoidal rules for 2D Schrodinger Equation. J. Math. Study, 48(1), 30-52.

Yakubu, D.G., Huoma, U.I. and Kwami, A.M. Symmetric Two-Step Runge-Kutta Collocation Methods for Stiff Systems of Ordinary Differential Equations, Journal of the Nigerian Mathematical Society, 2014, 33: 185-204.,


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MAYFEB Journal of Mathematics 
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MAYFEB TECHNOLOGY DEVELOPMENT
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