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UPSI Digital Repository (UDRep)
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| Abstract : Universiti Pendidikan Sultan Idris |
| This research is aimed to produce an efficient implementation technique for the 2-stage
(G2) and 3-stage (G3) implicit Runge-Kutta Gauss methods in solving mathematical stiff problems.
Both methods are constructed by using Maple software and have been implemented by using Matlab
software numerically. This research applied four imple- mentation strategies which are full Newton
without compensated summation (FNWSN), full Newton with compensated summation (FNCS), simplified
Newton without com- pensated summation (SNWCS) and simplified Newton with compensated summation
(SNCS). Comparison have been done with the implementations of Hairer and Wanner scheme, Cooper and
Butcher scheme, and González scheme. Results for stiff test prob- lems showed that SNCS is the most
efficient technique in solving some real life mathe- matical problems such as the Kepler,
Oregonator, Van der Pol, HIRES and Brusselator problems. According to the numerical results, the
implementation of G2 using SNCS by the Hairer and Wanner scheme is the most efficient technique for
solving Kepler and Brusselator problems, while SNCS by the González scheme is the most efficient
technique for solving other problems. On the contrary for G3, SNCS by the Hairer and Wanner scheme
gives the most efficient technique for solving Kepler and Van der Pol problems, while SNCS by the
González scheme gives the most efficient technique for solving other problems. In conclusion, for
both G2 and G3 methods, SNCS plays an important role to improve the efficiency of implicit
Runge-Kutta Gauss methods in solv- ing mathematical stiff problems. As for the implications, the
implementation technique used in this research can be extended during tertiary education on the
subject numer- ical ordinary differential equations that focusses on implementation schemes by
other
researchers as well as to some other implicit Runge-Kutta methods.
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| References |
Agam, S. A. and Yahaya, Y. A. (2014). A highly efficient implicit Runge-Kutta method for first order ordinary differential equations. African Journal of Mathematics and Computer Science Research, 7(5): 55–60.
Bashforth, F. (1883). An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluid: with an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, by JC Adams. University Press.
Berghe, G. V. and Van, D. M. (2011). Symplectic exponentially-fitted four-stage Runge- Kutta methods of the Gauss type. Numerical Algorithms, 56(4): 591–608.
Boom, P. D. and Zingg, D. W. (2015). Investigation of efficient high-order implicit Runge-Kutta methods based on generalized summation by parts operators. 22nd AIAA Computational Fluid Dynamics Conference, page 2757.
Brugnano, L., Caccia, G. F., and Iavernaro, F. (2014). Efficient implementation of Gauss collocation and Hamiltonian boundary value methods. Numerical Algorithms, 65(3): 633–650.
Brugnano, L., Iavernaro, F., and Trigiante, D. (2010). Hamiltonian boundary value methods (energy preserving discrete line integral methods). Journal Numerical Anal- ysis, Industrial and Applied Mathematics, 5(1-2): 17–37.
Butcher, J. C. (1963). Coefficients for the study of Runge-Kutta integration processes. Journal of the Australian Mathematical Society, 3(2): 185–201.
Butcher, J. C. (1964a). Implicit Runge-Kutta processes. Mathematics of Computation, 18(85): 50–64.
Butcher, J. C. (1964b). On Runge-Kutta processes of high order. Journal of the Aus- tralian Mathematical Society, 4(2): 179–194.
Butcher, J. C. (1976). On the implementation of implicit Runge-Kutta methods. BIT Numerical Mathematics, 16(3): 237–240.
Butcher, J. C. (1987). The numerical analysis of ordinary differential equations: Runge- Kutta and general linear methods. Australia: John Wiley and Sons Inc.
Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied Numerical Mathe- matics, 20(3): 247–260.
Butcher, J. C. (2016). Numerical methods for ordinary differential equations. Australia: John Wiley & Sons.
Butcher, J. C. and Hojjati, G. (2005). Second derivative methods with Runge-Kutta stability. Numerical Algorithms, 40(4): 415–429.
Calvo, M., Franco, J. M., Montijano, J. I., and Rández, L. (2009). Sixth-order symmetric and symplectic exponentially fitted Runge-Kutta methods of the Gauss type. Journal of Computational and Applied Mathematics, 223(1): 387–398.
Cash, J. R. (1975). A class of implicit Runge-Kutta methods for the numerical integra- tion of stiff ordinary differential equations. Journal of the ACM, 22(4): 504–511.
Chan, R. P. K. (1990). On symmetric Runge-Kutta methods of high order. Computing, 45(4): 301–309.
Chan, R. P. K. and Gorgey, A. (2013). Active and passive symmetrization of Runge- Kutta Gauss methods. Applied Numerical Mathematics, 67:64–77.
Cong, N. h. (1994). Parallel iteration of symmetric Runge-Kutta methods for nonstiff initial-value problems. Journal of Computational and Applied Mathematics, 51(1): 117–125.
Cooper, G. J. and Butcher, J. C. (1983). An iteration scheme for implicit Runge-Kutta methods. IMA Journal of Numerical Analysis, 3(2): 127–140.
Cooper, G. J. and Verner, J. H. (1972). Some explicit Runge-Kutta methods of high order. SIAM Journal on Numerical Analysis, 9(3): 389–405.
Curtis, A. R. (1970). An eighth order Runge-Kutta process with eleven function evalu- ations per step. Numerische Mathematik, 16(3): 268–277.
Curtis, A. R. (1975). High-order explicit Runge-Kutta formulae, their uses, and limita- tions. IMA Journal of Applied Mathematics, 16(1): 35–52.
Enright, W. H., Hull, T. E., and Lindberg, B. (1975). Comparing numerical methods for stiff systems of ODE. BIT Numerical Mathematics, 15(1): 10–48.
Faou, E., Hairer, E., and Pham, T. L. (2004). Energy conservation with non-symplectic methods: examples and counter-examples. BIT Numerical Mathematics, 44(4): 699– 709.
Gill, S. (1951). A process for the step-by-step integration of differential equations in an automatic digital computing machine. Mathematical Proceedings of the Cambridge Philosophical Society, 96-108.
González-Pinto, S., González-Concepción, C., and Montijano, J. I. (1994). Iterative schemes for Gauss methods. Computers and Mathematics with Applications, 27(7): 67–81.
González-Pinto, S., Montijano, J. I., and Rández, L. (1995). Iterative schemes for three- stage implicit Runge-Kutta methods. Applied Numerical Mathematics, 17(4): 363– 382.
Gorgey, A. (2012). Extrapolation of symmetrized Runge-Kutta methods. PhD thesis, ResearchSpace@ Auckland.
Gorgey, A. and Hafizul, M. (2017). Efficiency of Runge-Kutta methods in solving Kepler problem. AIP Conference Proceedings, 020016.
Hafizul, M. and Gorgey, A. (2018). Investigation on the most efficient ways to solve the implicit equations for Gauss methods in the constant stepsize setting. Applied Mathematical Sciences, 12(2): 93–103.
Hairer, E. (1978). A Runge-Kutta method of order 10. IMA Journal of Applied Mathe- matics, 21(1): 47–59.
Hairer, E., McLachlan, R. I., and Razakarivony, A. (2008). Achieving Brouwer’s law with implicit Runge-Kutta methods. BIT Numerical Mathematics, 48(2): 231–243.
Hairer, E. and Wanner, G. (1999). Stiff differential equations solved by Radau methods. Journal of Computational and Applied Mathematics, 111(1): 93–111.
Hitchens, F. (2015). Propeller Aerodynamics: The history, Aerodynamics & Operation of Aircraft Propellers. United Kingdom: Andrews UK Limited.
Huen, K. (1900). Neue methode zur approximativen integration der differentialge- ichungen einer unabhngigen variablen. Zeitschrift für angewandte Mathematik und Physik, 45: 23–38.
Huta, A. (1965). Une amtlioration de ia mtthode de Runge-Kutta-Nystrtim pour la rtsolution nurntrique des, quations diffentielles du premier ordre. Acta Math. Univ. Comenian, 1: 21–24.
Iserles, A. (2009). A first course in the numerical analysis of differential equations. Number 44. United State of America: Cambridge University press.
Kadoura, A., Sun, S., and Salama, A. (2014). Accelerating Monte Carlo molecular sim- ulations by reweighting and reconstructing Markov chains: Extrapolation of canon- ical ensemble averages and second derivatives to different temperature and density conditions. Journal of Computational Physics, 270: 70–85.
Kajanthan, S. and Vigneswaran, R. (2014). Some efficient implementation schemes for implicit Runge-Kutta methods. International Journal of Pure and Applied Mathe- matics, 93(4): 525–540.
Kulikov, G. Y. (2015). Embedded symmetric nested implicit Runge-Kutta methods of Gauss and Lobatto types for solving stiff ordinary differential equations and Hamilto- nian systems. Computational Mathematics and Mathematical Physics, 55(6): 983– 1003.
Kuntzmann, J. (1961). Neuere entwicklungen der methode von Runge und Kutta. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 41(S1).
Kutta, W. (1901). Beitrag zur näheruengweisen integration totaler differentialgleichun- gen.
Liniger, W. and Willoughby, R. A. (1970). Efficient integration methods for stiff systems of ordinary differential equations. SIAM Journal on Numerical Analysis, 7(1): 47– 66.
Nazari, F., Mohammadian, A., Charron, M., and Zadra, A. (2014). Optimal high-order diagonally-implicit Runge-Kutta schemes for nonlinear diffusive systems on atmo- spheric boundary layer. Journal of Computational Physics, 271: 118–130.
Prothero, A. and Robinson, A. (1974). On the stability and accuracy of one-step meth- ods for solving stiff systems of ordinary differential equations. Mathematics of Com- putation, 28(125): 145–162.
Rechenberg, H. (2001). The historical development of quantum theory, volume 1. United State of America: Springer Science & Business Media.
Runge, C. (1895). Über die numerische auflösung von differentialgleichungen. Mathe- matische Annalen, 46(2): 167–178.
Sanderse, B. and Koren, B. (2012). Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations. Journal of Computational Physics, 231(8): 3041–3063.
Shampine, L. F. (1980). Implementation of implicit formulas for the solution of ODEs. SIAM Journal on Scientific and Statistical Computing, 1(1): 103–118.
Shampine, L. F. (1984). Stability of explicit Runge-Kutta methods. Computers and Mathematics with Applications, 10(6): 419–432.
Shampine, L. F. and Gear, C. W. (1979). A user’s view of solving stiff ordinary differ- ential equations. SIAM review, 21(1): 1–17.
Skvortsov, L. M. and Kozlov, O. S. (2014). Efficient implementation of diagonally implicit Runge-Kutta methods. Mathematical Models and Computer Simulations, 6(4): 415–424.
Swart, D., Jacques, J. B., and Lioen, W. M. (1998). Collecting real-life problems to test solvers for implicit differential equations. CWI Quarterly, 11(1): 83–100.
Wanner, G. and Hairer, E. (1991). Solving ordinary differential equations II, volume 1. New York: Springer-Verlag, Berlin.
Williams, G. (2017). Linear algebra with applications. United State of America: Jones & Bartlett Learning.
Xie, D. (2011). An improved approximate Newton method for implicit Runge-Kutta formulas. Journal of Computational and Applied Mathematics, 235(17): 5249–5258.
Zhang, H., Sandu, A., and Tranquilli, P. (2015). Application of approximate matrix factorization to high order linearly implicit Runge-Kutta methods. Journal of Com- putational and Applied Mathematics, 286: 196–210.
Zhu, B., Hu, Z., Tang, Y., and Zhang, R. (2016). Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields. International Journal of Mode, Simulation, and Scientific Computing, 1658.
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