×

Nichtlineare Stabilität und Phasenuntersuchung adaptiver Nyström- Runge-Kutta-Methoden (Nonlinear stability and phase analysis for adaptive Nyström-Runge-Kutta methods). (German) Zbl 0569.65054

The stability of adaptive Nyström-Runge-Kutta procedures is studied for a wide class of nonlinear stiff systems of second order differential equations. We show that for a large class of semi-discrete hyperbolic and parabolic problems the restriction of the stepsize is not due to the stiffness of the differential equation. Furthermore we use the scalar test equation \(y''=-\omega^ 2y+q\cdot e^{iv(t-t_ 0)}\) to derive conditions which ensure that the numerical forced oscillation is in phase with the analytical forced oscillation. The order of adaptive Nyström- Runge-Kutta methods (with a stability-matrix based on a diagonal Padé- approximation) for which the forced oscillation is in phase with its analytical counterpart cannot be greater than two. This barrier of order is not true for r-stage implicit Nyström methods of order \(p=2r\).

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chawla, M. M., Sharma, S. R.: Intervals of periodicity and absolute stability of explicit Nyström methods fory”=f(x,y). BIT21, 455–464 (1981). · Zbl 0469.65048 · doi:10.1007/BF01932842
[2] Gladwell, I., Thomas, R. M.: Damping and phase analysis for some methods for solving secondorder ordinary differential equations. Int. J. num. Meth. Engng.19, 495–503 (1983). · Zbl 0513.65053 · doi:10.1002/nme.1620190404
[3] Hairer, E.: Unconditionally stable methods for second order differential equations. Numer. Math.32, 373–379 (1979). · Zbl 0393.65035 · doi:10.1007/BF01401041
[4] Houwen, van der P. J.: Stabilized Runge-Kutta methods for second order differential equations without first derivatives. SIAM J. Numer. Anal.16, 523–537 (1979). · Zbl 0415.65037 · doi:10.1137/0716040
[5] Jeltsch, R., Nevanlinna, O.: Stability of explicit time discretizations for solving initial value problems. Numer. Math.37, 61–91 (1981). · Zbl 0457.65054 · doi:10.1007/BF01396187
[6] Jeltsch, R., Nevanlinna, O.: Stability of semidiscretizations of hyperbolic problems. RWTH Aachen, Inst. f. Geometrie u. Prakt. Math., Bericht Nr 16 (1982). · Zbl 0532.65065
[7] Strehmel, K., Weiner, R.: Adaptive Nyström-Runge-Kutta-Methoden für gewohnliche Differentialgleichungssysteme zweiter Ordnung. Computing30, 35–47 (1983). · Zbl 0498.65038 · doi:10.1007/BF02253294
[8] Kramarz, L.: Stability of collocation methods for the numerical solution ofy”=f(x, y). BIT20, 215–222 (1980). · Zbl 0425.65043 · doi:10.1007/BF01933194
[9] Thomas, R. M.: Phase properties of high order, almostP-stable formulae. BIT24, 225–238 (1984). · Zbl 0569.65052 · doi:10.1007/BF01937488
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.