New extrapolation methods for initial value problems in ordinary differential equations. (English) Zbl 0962.65059
This paper is concerned with the numerical solution of initial value problems in ordinary differential equations by means of extrapolation methods based on rational one-step methods. Starting with one-step nonlinear rational methods which use high order derivatives, the authors propose both polynomial and rational extrapolation based on these methods. It is shown that the proposed methods are A-stable. Finally, some numerical results for the nonlinear scalar equation \( y' = 1 + y^2 \) when extrapolation is based on the inverse explicit Euler method are presented.
Reviewer: Manuel Calvo (Zaragoza)
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 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 |
Keywords:
initial value problems; inverse explicit Euler method; rational extrapolation; \(A\)-stability; one-step methods; numerical examplesReferences:
| [1] | DOI: 10.1007/BF02165234 · Zbl 0135.37901 · doi:10.1007/BF02165234 |
| [2] | Butcher J.C., The Numerical Analysis of ODEs: RKM and GLM (1987) · Zbl 0616.65072 |
| [3] | Cash J.R., SIAM J. Numer. Analysis 18 (1981) |
| [4] | Curtis A.R., The state of the art in Numerical Analysis (1987) |
| [5] | David R.H., Experiments in Computational Matrix Algebra (1988) · Zbl 0668.65023 |
| [6] | DOI: 10.1007/BF01418332 · Zbl 0543.65049 · doi:10.1007/BF01418332 |
| [7] | DOI: 10.1137/1027140 · Zbl 0602.65047 · doi:10.1137/1027140 |
| [8] | Eikemo M.S., Computing and Visualisation in Science 1 pp 1– |
| [9] | DOI: 10.1137/0711029 · Zbl 0249.65055 · doi:10.1137/0711029 |
| [10] | DOI: 10.1007/BF01932994 · Zbl 0301.65040 · doi:10.1007/BF01932994 |
| [11] | Evans D.J., Computational Mathematics 1 (1987) |
| [12] | DOI: 10.1016/0898-1221(82)90046-3 · Zbl 0483.65047 · doi:10.1016/0898-1221(82)90046-3 |
| [13] | Fatunla S.O., Journal Computer Maths with Applications 12 pp 109– (1986) |
| [14] | Fatunla S.O., Numerical Methods for IVPs in ODEs (1988) |
| [15] | Fatunla S.O., ABACUS 19 pp 121– (1990) |
| [16] | Fatunla S.O., Scientific Computing pp 68– (1994) |
| [17] | Fatunla S.O., Scientific Computing pp 46– (1994) |
| [18] | DOI: 10.1090/S0025-5718-1980-0559191-X · doi:10.1090/S0025-5718-1980-0559191-X |
| [19] | Gander W., Solving Problems in Scientific Computing Using MAPLE andMATLAB (1998) |
| [20] | Gear, C.W. The Automatic Integration of Stiff ODEs. Proc. IFIP Congress. Amsterdam. Vol. 1, pp.187–194. North-Holland. |
| [21] | DOI: 10.1145/362566.362573 · doi:10.1145/362566.362573 |
| [22] | Gragg W.B., SIAM J. Numer. Analysis 2 pp 384– (1965) |
| [23] | Hackbusch W., Computing and Visualisation in Science 1 pp 15– (1997) |
| [24] | Henrici P., Discrete Variable Methods in ODEs (1962) · Zbl 0112.34901 |
| [25] | Ikhile M.N.O., Ph.D. Thesis (1998) |
| [26] | Jeltsch R., Math. Comp 32 pp 1108– (1978) |
| [27] | DOI: 10.1007/BF01400542 · Zbl 0493.65038 · doi:10.1007/BF01400542 |
| [28] | DOI: 10.1007/BF01396187 · Zbl 0457.65054 · doi:10.1007/BF01396187 |
| [29] | Lambert J.D., Math. Comp 19 pp 456– (1965) |
| [30] | Lambert J.D., Computational Methods in ODEs (1973) |
| [31] | Lambert, J.C. Nonlinear Methods for Stiff Systems of ODEs. Conference on the Numerical Solution of ODEs Dundee. pp.77–88. |
| [32] | DOI: 10.1016/0898-1221(75)90003-6 · Zbl 0329.65049 · doi:10.1016/0898-1221(75)90003-6 |
| [33] | DOI: 10.1080/00207169608804537 · Zbl 0873.65081 · doi:10.1080/00207169608804537 |
| [34] | Otunta F.O., Intern. Journal of Computer Mathematics 62 (1998) |
| [35] | DOI: 10.1137/0727087 · Zbl 0727.65067 · doi:10.1137/0727087 |
| [36] | DOI: 10.1016/0898-1221(87)90004-6 · Zbl 0627.65088 · doi:10.1016/0898-1221(87)90004-6 |
| [37] | Wanner G., The state of the art in Numerical Analysis (1987) |
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.
