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

##### MSC:
 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems 34A34 Nonlinear ordinary differential equations and systems, general theory
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##### References:
  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  Burder, J., Linearly implicit runge – kutta methods based on implicit runge – kutta methods, Appl. numer. math., 13, 33-40, (1993) · Zbl 0790.65059  Butcher, J.; Chartier, P., The effective order of singly-implicit runge – kutta methods, Numer. algorithms, 20, 269-284, (1999) · Zbl 0936.65089  Cahlon, B., Numerical methods for initial value problems, Appl. numer. math., 5, 399-407, (1989) · Zbl 0689.65045  Cash, J., High order p-stable formulae for periodic initial value problems, Numer. math., 37, 355-370, (1981) · Zbl 0488.65029  Cash, J., Efficient p-stable methods for periodic initial value problems, Bit, 24, 248-252, (1984) · Zbl 0558.65053  Chawla, M., Unconditionally stable noumerov-type methods for second order differential equations, Bit, 23, 541-552, (1983) · Zbl 0523.65055  Cooper, J.; Butcher, J., An iterative scheme for implicit runge – kutta methods, IMA J. numer. anal., 3, 127-140, (1983) · Zbl 0525.65052  Dahlquist, G., On accuracy and unconditional stability of linear multistep methods for second order differential equations, Bit, 18, 133-136, (1978) · Zbl 0378.65043  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  Ehle, B.; Picel, Z., Two parameter arbitrary order exponential approximations for stiff equations, Math. comput., 29, 501-511, (1975) · Zbl 0302.65059  Fairweather, G., A note on the efficient implementation of certain pade’ methods of linear parabolic problems, Bit, 18, 101-109, (1978) · Zbl 0384.65036  I. Galdwell, J. Wang, Iterations and predictors for second order systems, in: W.F. Ames (Eds.), Proceedings, 14th IMACS World Congress on Comput. and Appl. Math., Georgia, vol. 3, 1994, pp. 1267-1270.  Lambert, J., Computational methods in ordinary differential equations, (1973), Wiley London · Zbl 0258.65069  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  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  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  Serbin, M., On factoring a class of complex symmetric matrices without pivoting, Math. comput., 35, 1231-1234, (1980) · Zbl 0463.65016  Shampine, L., Implementation of implicit formulas for the solution of ODE’s, SIAM J. sci. stat. comput., 1, 103-118, (1980) · Zbl 0463.65050  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  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  Xiao, A., Order results for algebraically stable mono-implicit runge – kutta methods, J. comput. appl. math., 17, 639-644, (1999) · Zbl 0949.65075
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