Efficiency of Extrapolated Runge-Kutta Methods in Solving Linear and Nonlinear Problems
Kecekapan Extrapolasi Kaedah Runge-Kutta dalam Menyelesaikan Masalah Linear dan Tak Linear
Extrapolation involves taking a certain linear combination of the numerical solutions of a base method applied with different stepsizes to obtain greater accuracy. This linear combination is done to eliminate the leading error term. The technique of extrapolation (passive and active) in accelerating convergence has been used successfully in numerical solution of ordinary differential equations. In this study, symmetric Runge-Kutta methods for solving linear and nonlinear stiff problem are considered. Symmetric methods admit asymptotic error expansion in even powers of the stepsize and are therefore of special interest because successive extrapolations can increase the order by two at time. Two ways of applying extrapolation are considered such as the active and the passive. It is interesting to know which modes of applying extrapolation are the most efficient when applied with symmetric methods. Results of numerical experiments are given which show the efficiency of the rational and polynomial extrapolated Implicit Midpoint Rule (IMR) and the Implicit Trapezoidal Rule (ITR) in solving Chemistry and Chemical Reaction problems. Numerical results show that in both types of extrapolation, the passive mode is considered to be the most efficient. The results also show that extrapolation with smoothing gives better results than without smoothing.
Cardone, A., Jackiewicz, Z., Sandu, A., & Zhangx, H. (2014). Extrapolation based implicit-explicit general linear methods. NumerAlgor, 65, 377-399.
Chan, R. P. K., & Gorgey, A. (2011). Order-4 symmetrized Runge-Kutta methods for stiff problems. Journal of Quality Measurement and Analysis, 7(1), 53-66.
Chan, R. P. K. (1993). Generalized symmetric Runge-Kutta methods. Computing, 50(1), 31-49.
Chan, R. P. K., & Gorgey, A. (2013). Active and passive symmetrization of Runge-Kutta Gauss methods. Applied Numerical Mathematics, 67, 64-77.
Faragó, I., Havasi, Á., & Zlatev, Z. (2010). Efficient implementation of stable Richardson Extrapolation algorithms. Computers and Mathematics with Applications, 60, 2309–2325.
Faragó, I., Havasi, Á., & Zlatev, Z. (2013). The convergence of diagonally implicit Runge– Kutta methods combined with Richardson Extrapolation. Computers and Mathematics with Applications, 65, 395–401.
Gorgey, A. (2012). Extrapolation of symmetrized Runge-Kutta methods (unpublished PhD Thesis). University of Auckland, New Zealand.
Gorgey, A. (2014). Extrapolation of Boundary Value Problems. Australian Journal of Basic and Applied Sciences, 8(11), 23-29.
Gragg, W. B. (1965). On extrapolation algorithm for ordinary initial value problems. SIAM J. Numer. Anal., 2, 384 –403.
Han, G., & Wang, R. (2002). Richardson extrapolation of iterated discrete Galerkin solution for twodimensional Fredholm integral equations. Journal of Computational and Applied Mathematics, 139, 49-63.
Hull, T. E., Enright, W. H., Fellen, B. M., & Sedgwick, A. E. (1972). Comparing numerical methods for ordinary differential equations. SIAM Journal of Numerical Analysis, 9(4), 603 – 637.
Muyakazi, J. B., & Patidar, K.C. (2008). On Richardson extrapolation for fitted operator finite difference. Applied Mathematics and computation, 201, 465-480.
doi: 10.1016/j. amc.2007.12.035
Prothero, A., & Robinson, A. (1974). On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comp., 28, 145 – 162.
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.
Shieh, D. S. S., Chang, Y., & Carmichael, G. R. (1988). The evaluation of numerical techniques for solution of stiff ordinary differential equations arising from chemical kinetic problems. Enviromental Software, 3 (1), 28 – 38.
Zlatev, Z., Georgiev, K., & Dimov, I. (2014). Studying absolute stability properties of the Richardson Extrapolation combined with explicit Runge–Kutta methods. Computers and Mathematics with Applications, 67, 2294–2307.
Zhongying, C., Guaoqiang, L., & Gnaneshwar, N. (2009). Richardson extrapolation of iterated discrete projection methods for eigenvalue approximation. Journal of Computational and Applied Mathematics, 223, 48-61.