Explicit Nordsieck methods with quadratic stability. (English) Zbl 1247.65104
The authors present the construction of explicit Nordsieck methods with \(s\) stages of order \(p = s-1\) and stage order \(q = p\) with inherent quadratic stability and quadratic stability with large regions of absolute stability for ordinary differential equations \(y'=f(y),\;t\in [t_0,T]\), \(y(t_0)=y_0\), where \(f:\mathbb R^m\to\mathbb R^m\) is continuous. Stability regions of these methods could be compared favourably with stability regions of corresponding general linear methods of the same order with inherent Runge-Kutta stability.
Reviewer: Narahari Parhi (Bhubaneswar)
MSC:
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
quadratic stability; inherent quadratic stability; general linear methods; order conditions; comparison of methods; explicit Nordsieck method; inherent Runge-Kutta stabilityReferences:
| [1] | Bartoszewski, Z., Jackiewicz, Z.: Construction of two-step Runge–Kutta methods of high order for ordinary differential equations. Numer. Algorithms 18(1), 51–70 (1998) · Zbl 0916.65083 · doi:10.1023/A:1019157029031 |
| [2] | Bartoszewski, Z., Jackiewicz, Z.: Nordsieck representation of two-step Runge–Kutta methods for ordinary differential equations. Appl. Numer. Math. 53(2–4), 149–163 (2005) · Zbl 1086.65069 · doi:10.1016/j.apnum.2004.08.010 |
| [3] | Butcher, J.C.: Diagonally-implicit multi-stage integration methods. Appl. Numer. Math. 11(5), 347–363 (1993) · Zbl 0773.65046 · doi:10.1016/0168-9274(93)90059-Z |
| [4] | Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2003) · Zbl 1040.65057 |
| [5] | Butcher, J.C., Jackiewicz, Z.: Diagonally implicit general linear methods for ordinary differential equations. BIT 33(3), 452–472 (1993) · Zbl 0795.65043 · doi:10.1007/BF01990528 |
| [6] | Butcher, J.C., Jackiewicz, Z.: Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. Appl. Numer. Math. 27, 1–12 (1996) · Zbl 0933.65080 · doi:10.1016/S0168-9274(97)00109-8 |
| [7] | Butcher, J.C., Jackiewicz, Z.: Construction of general linear methods with Runge–Kutta stability properties. Numer. Algorithms 36(1), 53–72 (2004) · Zbl 1055.65083 · doi:10.1023/B:NUMA.0000027738.54515.50 |
| [8] | Butcher, J.C., Jackiewicz, Z., Wright, W.M.: Error propagation of general linear methods for ordinary differential equations. J. Complexity 23(4–6), 560–580 (2007) · Zbl 1131.65068 · doi:10.1016/j.jco.2007.01.009 |
| [9] | Butcher, J.C., Wright, W.M.: The construction of practical general linear methods. BIT 43(4), 695–721 (2003) · Zbl 1046.65054 · doi:10.1023/B:BITN.0000009952.71388.23 |
| [10] | Chollom, J., Jackiewicz, Z.: Construction of two-step Runge–Kutta methods with large regions of absolute stability. J. Comput. Appl. Math. 157(1), 125–137 (2003) · Zbl 1024.65054 · doi:10.1016/S0377-0427(03)00382-0 |
| [11] | Conte, D., D’Ambrosio, R., Jackiewicz, Z.: Two-step Runge–Kutta methods with quadratic stability functions. J. Sci. Comput. 44(2), 191–218 (2010) · Zbl 1203.65107 · doi:10.1007/s10915-010-9378-x |
| [12] | D’Ambrosio, R.: Two-step collocation methods for ordinary differential equations. Doctoral thesis, University of Salerno, Fisciano (Salerno), Italy (2010) |
| [13] | D’Ambrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for ordinary differential equations. Numer. Algorithms 53(2–3), 195–217 (2010) · Zbl 1186.65107 · doi:10.1007/s11075-009-9280-5 |
| [14] | D’Ambrosio, R., Jackiewicz, Z.: Continuous two-step Runge–Kutta methods for ordinary differential equations. Numer. Algorithms 54(2), 169–193 (2010) · Zbl 1191.65092 · doi:10.1007/s11075-009-9329-5 |
| [15] | Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations. I. In: Springer Series in Computational Mathematics, vol. 8, 2nd edn. Springer, Berlin (1993). Nonstiff problems · Zbl 0789.65048 |
| [16] | Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Hoboken (2009) · Zbl 1211.65095 |
| [17] | Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge–Kutta methods for ordinary differential equations. SIAM J. Numer. Anal. 32(5), 1390–1427 (1995) · Zbl 0837.65074 · doi:10.1137/0732064 |
| [18] | Jackiewicz, Z., Tracogna, S.: Variable stepsize continuous two-step Runge–Kutta methods for ordinary differential equations. Numer. Algorithms 12(3–4), 347–368 (1996) · Zbl 0869.65050 · doi:10.1007/BF02142812 |
| [19] | Jackiewicz, Z., Verner, J.H.: Derivation and implementation of two-step Runge–Kutta pairs. Jpn. J. Indt. Appl. Math. 19(2), 227–248 (2002) · Zbl 1006.65075 · doi:10.1007/BF03167454 |
| [20] | Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28, 145–162 (1974) · Zbl 0309.65034 · doi:10.1090/S0025-5718-1974-0331793-2 |
| [21] | Wright, W.M.: Explicit general linear methods with inherent Runge–Kutta stability. Numer. Algorithms 31(1–4), 381–399 (2002). Numerical methods for ordinary differential equations (Auckland, 2001) · Zbl 1016.65049 |
| [22] | Wright, W.M.: General linear methods with inherent runge-kutta stability. Doctoral thesis, The University of Auckland, New Zealand (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. 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.
