×

zbMATH — the first resource for mathematics

Variable order and stepsize in general linear methods. (English) Zbl 1416.65187
Summary: This paper describes the implementation of a class of IRKS methods [W. M. Wright, General linear methods with Inherent Runge-Kutta stability. Auckland: The University of Auckland. (PhD Thesis) (2002)]. These GLM algorithms are practical with reliable error estimators [the second author and H. Podhaisky, Appl. Numer. Math. 56, No. 3–4, 345–357 (2006; Zbl 1089.65080)]. The current robust ODE solvers in variable stepsize as well as in variable-order mode are based upon heuristics. In this paper, we examine an optimisation approach, based on Euler-Lagrange theory [the second author, IMA J. Numer. Anal. 6, 433–438 (1986; Zbl 0615.65076); Computing 44, No. 3, 209–220 (1990; Zbl 0719.65059)] to control the stepsize as well as the order and implement the GLMs in an efficient manner. A set of nonstiff to mildly stiff problems have been used to investigate this approach in fixed-order and variable-order modes.
MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
Software:
RADAU
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahmad, S.: Efficient implementation of implicit Runge-Kutta methods. Masters thesis, The University of Auckland. This can be provided on demand by the author (sahm056@aucklanduni.ac.nz) (2011)
[2] Ahmad, S.: Design and construction of software for general linear methods, PhD thesis, Massey University, Auckland, https://mro.massey.ac.nz/handle/10179/9925(2016)
[3] Butcher, JC, Optimal order and stepsize sequences,, IMA J. Numer. Anal., 6, 433-438, (1986) · Zbl 0615.65076
[4] Butcher, JC, Order, stepsize and stiffness switching, Computing, 44, 209-220, (1990) · Zbl 0719.65059
[5] Butcher, JC, General linear methods for the parallel solution of ordinary differential equations, World Sci. Ser. Appl. Anal., 2, 99-111, (1993) · Zbl 0834.65059
[6] Butcher, JC, An introduction to DIMSIMs, Comput. Appl. Math., 14, 59-72, (1995) · Zbl 0838.65078
[7] Butcher, JC, General linear methods for stiff differential equations, BIT, 41, 240-264, (2001) · Zbl 0983.65085
[8] Butcher, J.C.: Numerical methods for ordinary differential equations. Wiley, Chichester (2016) · Zbl 1354.65004
[9] Butcher, JC, Thirty years of G-stability, BIT, 46, 479-489, (2006) · Zbl 1105.65085
[10] Butcher, J.C., Chartier, P., Jackiewicz, Z.: Experiments with variable – order type 1 DIMSIM code. Numer. Algorithms 22, 237- (1999) · Zbl 0958.65083
[11] Butcher, JC; Podhaisky, H., On error estimation in general linear methods for stiff ODEs, Appl. Numer. Math., 56, 345-357, (2006) · Zbl 1089.65080
[12] Butcher, JC; Jackiewicz, Z., A new approach to error estimation for general linear methods, Numer. Math., 95, 487-502, (2003) · Zbl 1032.65088
[13] Butcher, JC; Jackiewicz, Z.; Wright, WM, Error propagation for general linear methods for ordinary differential equations, J. Complexity, 23, 560-580, (2007) · Zbl 1131.65068
[14] Butcher, JC; Wright, WM, The construction of practical general linear methods, BIT, 43, 695-721, (2003) · Zbl 1046.65054
[15] Butcher, JC; Wright, WM, Application of doubly companion matrices, Appl. Numer. Math., 56, 358-373, (2006) · Zbl 1089.65065
[16] Gladwell, I.; Shampine, LF; Brankin, RW, Automatic selection for the initial stepsize for an ODE solver, J. Comput. Appl. Math., 18, 175-192, (1987) · Zbl 0623.65080
[17] Gustafsson, 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
[18] Hairer, E.; Wanner, G., Stiff differential equations solved by Radau methods, J. Comput. Appl. Math., 111, 93-111, (1999) · Zbl 0945.65080
[19] Hairer, E., Wanner, G.: Solving ordinary differential equations II. Springer, Berlin (1996) · Zbl 0859.65067
[20] Hull, TE; Enright, WH; Fellen, BM; Sedgwick, AE, Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal., 9, 603-637, (1972) · Zbl 0221.65115
[21] Jackiewicz, Z.: General linear methods for ordinary differential equations. Wiley, Chichester (2009) · Zbl 1211.65095
[22] Shampine, L.F., Gordon, M.K.: Using DE/STEP, INTRP to Solve Ordinary Differential Equations, United States: N. p. Web (1974)
[23] Wright, W.M.: General Linear Methods with Inherent Runge-Kutta Stability. PhD thesis, The University of Auckland (2002) · 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.