All Title Author
Keywords Abstract

The Explicit Fatunla’s Method for First-Order Stiff Systems of Scalar Ordinary Differential Equations: Application to Robertson Problem

DOI: 10.4236/oalib.1105291, PP. 1-15

Keywords: Stiff Differential Equations, Explicit Fatunla Method, Robertson Problem, Stability

Full-Text   Cite this paper   Add to My Lib


Ordinary differential equations (ODEs) are among the most important mathe-matical tools used in producing models in the physical sciences, biosciences, chemical sciences, engineering and many more fields. This has motivated re-searchers to provide efficient numerical methods for solving such equations. Most of these types of differential models are stiff, and suitable numerical methods have to be used to simulate the solutions. This paper starts with a sur-vey on the basic properties of stiff differential equations. Thereafter, we present the explicit one-step algorithm proposed by Fatunla to solve stiff systems of first-order scalar ODEs. As an illustrative example, we consider the Robertson problem (RP) which is known to be stiff. The results obtained with the explicit Fatunla method (EFM) are compared with those computed by the solver RADAU which is based on implicit Runge-Kutta methods. Our results are in good agreement with the latter ones.


[1]  Atkinson, K.E., Han, W. and Stewart, D. (2009) Numerical Solution of OrdinaryDifferential Equations. John Wiley & Sons, Hoboken, NJ.
[2]  Hairer, E. and Wanner, G. (1996) Solving Ordinary Differential Equations II, Stiff Differential-Algebraic Problems. 2nd Edition, Springer-Verlag, Berlin.
[3]  Niemeyer, K.E. and Sung, C.J. (2014) GPU-Based Parallel Integration of Large Numbers of Independent ODE Systems in Numerical Computations with GPUs. Chapter 8, 159-188.
[4]  Mazzia, F. and Iavernaro, F. (2003) Test Set for Initial Value Problem Solvers, Release 2.4.
[5]  Akinfenwa, O., Jator, S. and Yoa, N. (2011) An Eight Order Backward Differential Formula with Continuous Coefficients of Stiff Ordinary Differential Equations. International Journal of Mathematical and Computer Sciences, 7, 171-176.
[6]  Curtiss, C.F. and Hirschfelder, J.O. (1952) Integration of Stiff Equations. Proceedings of the National Academy of Sciences of the United States of America, 38, 235-243.
[7]  Butcher, J.C. (2016) Numerical Methods for Ordinary Differential Equation. 3rd Edition, John Wiley & Sons, Hoboken, NJ.
[8]  Sandu, A., Verwer, J.G., Blom, J.G., Spee, E.J., Carmichael, G.R. and Potra, F.A. (1997) Benchmarking Stiff ODE Solvers for Atmospheric Chemistry Problems II: Rosenbrock Solvers. Atmospheric Environment, 31, 3459-3479.
[9]  Kin, J. and Cho, S.Y. (1997) Computational Accuracy and Efficiency of the Time-Splitting Method in Solving Atmospheric Transport/Chemistry Equations. Atmospheric Environment, 31, 2215-2224.
[10]  Bonnard, B., Faubourg, L. and Trélat, E. (2006) Mécanique Céleste et Controle des Véhicules Spatiaux. Springer, Berlin.
[11]  Shampire, L.F. and Gear, C.W. (1979) A User’s View of Solving Stiff Ordinary Differential Equations. Society for Industrial and Applied Mathematics, 21, 1-17.
[12]  Ebady, A.M.N., Habib, H.M. and El-Zahar, E.R. (2012) A Fourth Order A-Stable Explicit One-Step Method for Solving Stiff Differential Systems Arising in Chemical Reactions. International Journal of Pure and Applied Mathematics, 81, 803-812.
[13]  Gupta, G.K., Sacks-Davis, R. and Tischer, P.E. (1985) A Review of Recent Developments in Solving ODEs. Computing Surveys, 17, 5-47.
[14]  Bjurel, G., Dahlquist, G.G., Lindberg, B., Linde, S. and Oden, L. (1970) Survey of Stiff Ordinary Differential Equations. Report NA 70.11, KTH Royal Institute of Technology, Stockholm, Sweden.
[15]  Fatunla, S.O. (1980) Numerical Integrators for Stiff and Highly Oscillatory Differential Equations. Mathematics of Computation, 34, 373-390.
[16]  Robertson, H.H. (1967) The Solution of a Set of Reaction Rate Equations. In: Walsh, J., Ed., Numerical Analysis: An introduction, Academic Press, Cambridge, Massachusetts, 178-182.
[17]  Shampire, L.F. (1994) Numerical Solution of Ordinary Differential Equations. Chapman & Hall, UK.
[18]  Edsberg, L. (1974) Integration Package for Chemical Kinetics. Plenum Press, Heidelberg, Germany, 81-94.
[19]  Gobbert, M.K. (1996) Rob-ertson’s Example for Stiff Differential Equations. Technical Report, Arizona State University, Tempe, AZ.
[20]  Frapiccini, A.L., et al. (2014) Explicit Schemes for Time Propagating Many-Body Wave Functions. Physical Review A, 89, Article ID: 023418.
[21]  Aiken, R.C. (1985) Stiff Computation. Oxford University Press, Oxford, England.
[22]  Moler, C.B. (2004) Numerical Computing with Matlab. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania.
[23]  Merriam-Webster (1989) Webster’s Ninth New Collegiate Dictionary. Merriam-Webster, Springfield, Massachusetts.
[24]  Hornby, A.S., Gatenby, E.V. and Wakefield, H. (1963) The Advanced Learner’s Dictionary of Current English. Oxford University Press, Oxford, England.
[25]  Little, W., Fowler, H.W., Coulson, J. and Onions, C.T. (1068) The Shorter Oxford English Dictionary on Historical Principles. 3rd Edition, Clarendon Press, Oxford England.
[26]  Dahlquist, G. (1963) A Special Stability Problem for Linear Multistep Methods. BIT Numerical Mathematics, 3, 27-43.
[27]  Jedrzejewski, F. (2005) Introduction aux Méthodes Numériques. Deuxième Edition, Springer-Verlag, Berlin.
[28]  Hairer, E. and Wanner, G. (1999) Stiff Differential Equations Solved by Randau Methods. Journal of Computational and Applied Mathematics, 111, 93-111.
[29]  Iserles, A. (2008) A First Course in the Numerical Analysis of Differen-tial Equations. 2nd Edition, Cambridge University Press, Cambridge, England.
[30]  Anake, T.A., Bishop, S.A., Adesanya, A.O. and Agarana, M.C. (2014) An -Stable Method for Solving Initial-Value Problems of Ordinary Differential Equations. Advances in Differential Equations and Control Processes, 13, 21-35.
[31]  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, 459, 797-815.
[32]  Lambert, J.D. (1991) Numerical Methods for Ordinary Differential Systems. The Ini-tial-Value Problems. John Wiley & Sons, Hoboken, NJ.
[33]  Sekar, S. and Nalini, M. (2015) Numerical Solution of Linear and Nonlinear Stiff Problems Using Adomian Decomposition Method. IOSR Journal of Mathematics, 11, 14-20.
[34]  Cartwiright, J.H.E. (1999) Nonlinear Stiffness, Lyapunov Exponents, and Attractor Dimension. Physics Letters A, 264, 298-302.
[35]  Gaffney, P.W. (1984) A Performance Evaluation of Some FORTRAN Subroutines for the Solution of Stiff Oscillatory ODEs. ACM Transactions on Mathematical Softwares, 10, 58-72.
[36]  Fatunla, S.O. (1998) Numerical Methods for Initial Value Problems in Ordinary Dif-ferential Equations. Academic Press, Cambridge, Massachusetts.
[37]  Lambert, J.D. (1980) Stiffness. In: Gradwell, I. and Sayers, D.K., Eds., Computational Techniques for Ordinary Differential Equations, Academic Press, Cambridge, Massachu-setts.
[38]  S?derlind, G., Jay, L. and Calvo, M. (2015) Stiffness 1952-2012: Sixty Years in Search of a Definition. BIT Numerical Mathematics, 55, 531-558.
[39]  S?derlind, G. (2006) The Logarithmic Norm. His-tory and Modern Theory. BIT Numerical Mathematics, 46, 631-652.
[40]  Madro?ero, J. and Piraux, B. (2009) Explicit Time-Propagation Method to Treat the Dynamics of Driven Complex Systems. Physical Review A, 80, Article ID: 033409.


comments powered by Disqus