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Accuracy and linear stability of RKN methods for solving second-order stiff problems. (English) Zbl 1161.65062
Summary: A general analysis of accuracy and linear stability of Runge-Kutta-Nyström (RKN) methods for solving second-order stiff problems is carried out. This analysis reveals that when components with large frequencies (stiff frequencies) and small amplitudes appear in the solution of the problem, the accuracy of an unconditionally stable RKN method can be seriously affected unless certain algebraic conditions are satisfied. Based on these algebraic conditions we derive new fourth-order A-stable diagonally implicit RKN (DIRKN) methods with different dispersion order and stage order. The numerical experiments carried out show the efficiency of the new methods when they are compared with other DIRKN codes proposed in the scientific literature for solving second-order stiff problems.

65L20Stability and convergence of numerical methods for ODE
65L05Initial value problems for ODE (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods
34A34Nonlinear ODE and systems, general
65L70Error bounds (numerical methods for ODE)
Full Text: DOI
[1] Alonso-Mallo, I.; Cano, B.; Moreta, M. J.: Stability of Runge -- Kutta -- Nyström methods, J. comput. Appl. math. 189, 120-131 (2006) · Zbl 1089.65076 · doi:10.1016/j.cam.2005.01.005
[2] Brusa, L.; Nigro, L.: A one-step method for direct integration of structural dynamic equations, Internat. J. Numer. methods eng. 15, 685-699 (1980) · Zbl 0426.65034 · doi:10.1002/nme.1620150506
[3] Cash, J. R.: High order, P-stable formulae for the numerical integration of periodic initial value problems, Numer. math. 37, 355-370 (1981) · Zbl 0488.65029 · doi:10.1007/BF01400315
[4] Chawla, M. M.; Rao, P. S.; Neta, B.: Two-step fourth order P-stable methods with phase-lag of order six for y″=$f(t,y)$, J. comput. Appl. math. 16, 233-236 (1986) · Zbl 0596.65047 · doi:10.1016/0377-0427(86)90094-4
[5] Chawla, M. M.; Sharma, S. R.: Intervals of periodicity and absolute stability of explicit Nyström methods, Bit 21, 455-464 (1981) · Zbl 0469.65048 · doi:10.1007/BF01932842
[6] Franco, J. M.; Gómez, I.: Fourth-order symmetric DIRK methods for periodic stiff problems, Numer. algorithms 32, 317-336 (2003) · Zbl 1058.65072 · doi:10.1023/A:1024077930017
[7] Franco, J. M.; Gómez, I.; Rández, L.: SDIRK methods for odes with oscillating solutions, J. comput. Appl. math. 81, 197-209 (1997) · Zbl 0887.65078 · doi:10.1016/S0377-0427(97)00056-3
[8] Franco, J. M.; Gómez, I.; Rández, L.: Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order, Numer. algorithms 26, 347-363 (2001) · Zbl 0974.65076 · doi:10.1023/A:1016629706668
[9] González-Pinto, S.; Pérez-Rodríguez, S.; Rojas-Bello, R.: Efficient iterations for Gauss methods on second order problems, J. comput. Appl. math. 189, 80-97 (2006) · Zbl 1086.65065 · doi:10.1016/j.cam.2006.05.014
[10] S. González-Pinto, R. Rojas-Bello, A code based on the two-stage Gauss method for second order problems, preprint in (http://pcmap.unizar.es/numerico/), ACM Trans. Math. Softw., submitted for publication
[11] Hairer, E.; Nørsett, S. P.; Wanner, G.: Solving ordinary differential equations I, nonstiff problems, (1993) · Zbl 0789.65048
[12] Hairer, E.; Wanner, G.: Solving ordinary differential equations II, stiff and differential-algebraic problems, (1991) · Zbl 0729.65051
[13] Papageorgiou, G.; Famelis, I. T.; Tsitouras, C.: A P-stable singly diagonally implicit Runge -- Kutta -- Nyström method, Numer. algorithms 17, 345-353 (1998) · Zbl 0939.65097 · doi:10.1023/A:1016644726305
[14] Sharp, P. W.; Fine, J. M.; Burrage, K.: Two-stage and three-stage diagonally implicit Runge -- Kutta -- Nyström methods of order three and four, IMA J. Numer. anal. 10, 489-504 (1990) · Zbl 0711.65057 · doi:10.1093/imanum/10.4.489
[15] Thomas, R. M.: Phase properties of high-order, almost P-stable formulae, Bit 24, 225-238 (1984) · Zbl 0569.65052 · doi:10.1007/BF01937488
[16] Van Der Houwen, P. J.; Sommeijer, B. P.: Explicit Runge -- Kutta ( -- Nyström) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. anal. 24, 595-617 (1987) · Zbl 0624.65058 · doi:10.1137/0724041
[17] Van Der Houwen, P. J.; Sommeijer, B. P.: Phase-lag analysis of implicit Runge -- Kutta methods, SIAM J. Numer. anal. 26, 214-229 (1989) · Zbl 0669.65055 · doi:10.1137/0726012
[18] Van Der Houwen, P. J.; Sommeijer, B. P.: Diagonally implicit Runge -- Kutta -- Nyström methods for oscillatory problems, SIAM J. Numer. anal. 26, 414-429 (1989) · Zbl 0676.65072 · doi:10.1137/0726023