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 |
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