Towards a code for nonstiff differential systems based on general linear methods with inherent Runge-Kutta stability. (English) Zbl 1405.65086
Summary: Various issues are discussed which are relevant to the development of the code for nonstiff differential systems based on a class of general linear methods with inherent Runge-Kutta stability.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
Keywords:
general linear methods; Nordsieck representation; inherent Runge-Kutta stability; adaptive stepsize and order selectionReferences:
| [1] | Butcher, J. C.; Chartier, P.; Jackiewicz, Z., Experiments with a variable-order type 1 DIMSIM code, Numer. Algorithms, 22, 237-261, (1999) · Zbl 0958.65083 |
| [2] | Butcher, J. C.; Jackiewicz, Z., A new approach to error estimation for general linear methods, Numer. Math., 95, 487-502, (2003) · Zbl 1032.65088 |
| [3] | Butcher, J. C.; Jackiewicz, Z., Construction of general linear methods with Runge-Kutta stability properties, Numer. Algorithms, 36, 53-72, (2004) · Zbl 1055.65083 |
| [4] | Butcher, J. C.; Jackiewicz, Z., Unconditionally stable general linear methods for ordinary differential equations, BIT, 44, 557-570, (2004) · Zbl 1066.65078 |
| [5] | Butcher, J. C.; Jackiewicz, Z.; Wright, W. M., Error propagation of general linear methods for ordinary differential equations, J. Complex., 23, 560-580, (2007) · Zbl 1131.65068 |
| [6] | Butcher, J. C.; Wright, W. M., The construction of practical general linear methods, BIT, 43, 695-721, (2003) · Zbl 1046.65054 |
| [7] | Gustafson, K.; Lundh, M.; Söderlind, G., A PI stepsize control for the numerical solution of ordinary differential equations, BIT, 28, 270-287, (1988) · Zbl 0645.65039 |
| [8] | Gustafsson, K., Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods, ACM Trans. Math. Softw., 17, 533-554, (1991) · Zbl 0900.65256 |
| [9] | Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, (1993), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0789.65048 |
| [10] | Hull, T. E.; Enright, W. H.; Fellen, B. M.; Sedgwick, A. E., Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal., 9, 603-637, (1972) · Zbl 0221.65115 |
| [11] | Jackiewicz, Z., Implementation of DIMSIMs for stiff differential systems, Appl. Numer. Math., 42, 251-267, (2002) · Zbl 1001.65082 |
| [12] | Jackiewicz, Z., General Linear Methods for Ordinary Differential Equations, (2009), John Wiley: John Wiley Hoboken, New Jersey · Zbl 1211.65095 |
| [13] | Shampine, L. F.; Gladwell, I.; Thompson, S., Solving ODEs with MATLAB, (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1079.65144 |
| [14] | Shampine, L. F.; Reichelt, M. W., The Matlab ODE suite, SIAM J. Sci. Comput., 18, 1-22, (1997) · Zbl 0868.65040 |
| [15] | Söderlind, G., The automatic control in numerical integration, CWI Quart., 11, 55-74, (1998) · Zbl 0922.65063 |
| [16] | Söderlind, G., Automatic control and adaptive time-stepping, Numer. Algorithms, 31, 281-310, (2002) · Zbl 1012.65080 |
| [17] | Wright, W., General Linear Methods with Inherent Runge-Kutta Stability, (2002), The University of Auckland: The University of Auckland New Zealand, Ph.D. thesis · Zbl 1016.65049 |
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.
