×
Attili, Basem S.; Furati, Khalid; Syam, Muhammed I.

An efficient implicit Runge-Kutta method for second order systems. (English) Zbl 1100.65062

Appl. Math. Comput. 178, No. 2, 229-238 (2006).
Summary: We consider the efficient implementation of a fourth order two stage implicit Runge-Kutta method to solve periodic second order initial value problems. To solve the resulting systems, we use the factorization of the discretized operator. Such proposed factorization involves both complex and real arithmetic. The latter case is considered here. The resulting system is efficient and small in size. It is one fourth the size of systems using normal implicit Runge-Kutta method. Numerical details and examples are also presented to demonstrate the efficiency of the method.

Cited in 14 Documents

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems

Keywords:

numerical examples; fourth order two stage implicit Runge-Kutta method; periodic second order initial value problems; factorization
PDF BibTeX XML Cite
\textit{B. S. Attili} et al., Appl. Math. Comput. 178, No. 2, 229--238 (2006; Zbl 1100.65062)
Full Text: DOI

References:

[1] Attili, B.; Elgindi, M.; Elgebeily, M., Initial value methods for the eigenelements of singular two-point boundary value problems, AJSE, 22, 2C, 67-77 (1997) · Zbl 0910.65058
[2] Burder, J., Linearly implicit Runge-Kutta methods based on implicit Runge-Kutta methods, Appl. Numer. Math., 13, 33-40 (1993) · Zbl 0790.65059
[3] Butcher, J.; Chartier, P., the effective order of singly-implicit Runge-Kutta methods, Numer. Algorithms, 20, 269-284 (1999) · Zbl 0936.65089
[4] Cahlon, B., Numerical methods for initial value problems, Appl. Numer. Math., 5, 399-407 (1989) · Zbl 0689.65045
[5] Cash, J., High order p-stable formulae for periodic initial value problems, Numer. Math., 37, 355-370 (1981) · Zbl 0488.65029
[6] Cash, J., Efficient p-stable methods for periodic initial value problems, BIT, 24, 248-252 (1984) · Zbl 0558.65053
[7] Chawla, M., Unconditionally stable Noumerov-type methods for second order differential equations, BIT, 23, 541-552 (1983) · Zbl 0523.65055
[8] Cooper, J.; Butcher, J., An iterative scheme for implicit Runge-Kutta methods, IMA J. Numer. Anal., 3, 127-140 (1983) · Zbl 0525.65052
[9] Dahlquist, G., On accuracy and unconditional stability of linear multistep methods for second order differential equations, BIT, 18, 133-136 (1978) · Zbl 0378.65043
[10] de-Swart, J.; Soderlind, G., On the construction of error estimators for implicit Runge-Kutta methods, J. Comput. Appl. Math., 86, 347-358 (1997) · Zbl 0897.65051
[11] Ehle, B.; Picel, Z., Two parameter arbitrary order exponential approximations for stiff equations, Math. Comput., 29, 501-511 (1975) · Zbl 0302.65059
[12] Fairweather, G., A note on the efficient implementation of certain Pade’ methods of linear parabolic problems, BIT, 18, 101-109 (1978) · Zbl 0384.65036
[14] Lambert, J., Computational Methods in Ordinary Differential Equations (1973), Wiley: Wiley London · Zbl 0258.65069
[15] Li, S. F.; Gan, S., A class of parallel multistep Runge-Kutta predictor-corrector algorithms, J. Numer. Methods Comput. Appl., 17, 1-11 (1995) · Zbl 0892.65048
[16] Liu, M.; Kraaijevanger, J., On the solvability of the systems of equations arising in implicit Runge-Kutta methods, Bit, 28, 825-838 (1988) · Zbl 0661.65078
[17] Olsson, H.; Soderlind, G., Stage value predictors and efficient newton iterations in implicit Runge-Kutta methods, SIAM J. Sci. Comput., 20, 185-202 (1999) · Zbl 0914.65078
[18] Serbin, M., On factoring a class of complex symmetric matrices without pivoting, Math. Comput., 35, 1231-1234 (1980) · Zbl 0463.65016
[19] Shampine, L., Implementation of implicit formulas for the solution of ODE’s, SIAM J. Sci. Stat. Comput., 1, 103-118 (1980) · Zbl 0463.65050
[20] Sharp, P.; Fine, J.; Burrage, K., Two-stage and three stage diagonally implicit Runge-Kutta Nystrom methods of order three and four, IMA J. Numer. Anal., 10, 489-504 (1990) · Zbl 0711.65057
[21] Voss, D.; Muir, P., Mono-implicit Runge-Kutta schemes for the parallel solutions of initial value ODE’s, J. Comput. Appl. Math., 102, 235-252 (1999) · Zbl 0945.65077
[22] Xiao, A., Order results for algebraically stable mono-implicit Runge-Kutta methods, J. Comput. Appl. Math., 17, 639-644 (1999) · Zbl 0949.65075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.