Difference schemes for nonlinear BVPs using Runge-Kutta IVP-solvers. (English) Zbl 1139.65057
Summary: Difference schemes for two-point boundary value problems for systems of first-order nonlinear ordinary differential equations are considered. It was shown in former papers of the authors that starting from the two-point exact difference scheme (EDS) one can derive a so-called truncated difference scheme (TDS) which a priori possesses an arbitrary given order of accuracy \({\mathcal O}(|h|^m)\) with respect to the maximal step size \(|h|\). This \(m\)-TDS represents a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid.
In he present paper, new efficient methods for the implementation of an \(m\)-TDS are discussed. Examples are given which illustrate the theorems proved in this paper.
In he present paper, new efficient methods for the implementation of an \(m\)-TDS are discussed. Examples are given which illustrate the theorems proved in this paper.
MSC:
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
Keywords:
numerical examples; difference schemes; two-point boundary value problems; systems of first-order nonlinear ordinary differential equations; truncated difference scheme; implementationSoftware:
RWPMReferences:
| [1] | Agarwal RP, Meehan M, O’Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge; 2001:x+170. · Zbl 0960.54027 · doi:10.1017/CBO9780511543005 |
| [2] | Ascher UM, Mattheij RMM, Russell RD: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall Series in Computational Mathematics. Prentice Hall, New Jersey; 1988:xxiv+595. · Zbl 0671.65063 |
| [3] | Deuflhard, P.; Bader, G.; Deuflhard, P. (ed.); Hairer, E. (ed.), Multiple shooting techniques revisited, 74-94 (1983), Massachusetts · Zbl 0527.65059 · doi:10.1007/978-1-4684-7324-7_6 |
| [4] | Hairer E, Nørsett SP, Wanner G: Solving Ordinary Differential Equations. I. Nonstiff Problems, Springer Series in Computational Mathematics. Volume 8. 2nd edition. Springer, Berlin; 1993:xvi+528. · Zbl 0789.65048 |
| [5] | Hermann M (Ed): Numerische Behandlung von Differentialgleichungen. III. Friedrich-Schiller-Universität, Jena; 1985:vi+150. |
| [6] | Hermann M: Numerik gewöhnlicher Differentialgleichungen. Oldenbourg, München; 2004. · Zbl 1097.65076 · doi:10.1524/9783486594980 |
| [7] | Hermann M, Kaiser D: RWPM: a software package of shooting methods for nonlinear two-point boundary value problems.Applied Numerical Mathematics 1993,13(1-3):103-108. · Zbl 0785.65089 · doi:10.1016/0168-9274(93)90134-D |
| [8] | Kutniv MV: Accurate three-point difference schemes for second-order monotone ordinary differential equations and their implementation.Computational Mathematics and Mathematical Physics 2000,40(3):368-382. translated from Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 40 (2000), no. 3, 387-401 · Zbl 0990.65087 |
| [9] | Kutniv MV: Three-point difference schemes of a high order of accuracy for systems of second-order nonlinear ordinary differential equations.Computational Mathematics and Mathematical Physics 2001,41(6):860-873. translated from Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 41 (2001), no. 6, 909-921 · Zbl 1029.65086 |
| [10] | Kutniv MV: High-order accurate three-point difference schemes for systems of second-order ordinary differential equations with a monotone operator.Computational Mathematics and Mathematical Physics 2002,42(5):754-738. translated from Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 42 (2002)), no. 5, 754-768 · Zbl 1060.65082 |
| [11] | Kutniv MV: Three-point difference schemes of high accuracy order for systems of nonlinear second order ordinary differential equations with the boundary conditions of the third art.Visnyk Lvivskogo Universytetu, Seria Prykladna Mathematyka ta Informatyka 2002, 4: 61-66. · Zbl 1063.65556 |
| [12] | Kutniv MV: Modified three-point difference schemes of high-accuracy order for second order nonlinear ordinary differential equations.Computational Methods in Applied Mathematics 2003,3(2):287-312. · Zbl 1038.65067 |
| [13] | Kutniv MV, Makarov VL, Samarskiĭ AA: Accurate three-point difference schemes for second-order nonlinear ordinary differential equations and their implementation.Computational Mathematics and Mathematical Physics 1999,39(1):40-55. translated from Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 39 (1999), no. 1, 45-60 · Zbl 0970.65085 |
| [14] | Makarov VL, Gavrilyuk IP, Kutniv MV, Hermann M: A two-point difference scheme of an arbitrary order of accuracy for BVPs for systems of first order nonlinear ODEs.Computational Methods in Applied Mathematics 2004,4(4):464-493. · Zbl 1068.65102 · doi:10.2478/cmam-2004-0026 |
| [15] | Makarov VL, Guminskiĭ VV: A three-point difference scheme of a high order of accuracy for a system of second-order ordinary differential equations (the nonselfadjoint case).Differentsial’nye Uravneniya 1994,30(3):493-502, 550. · Zbl 0823.65078 |
| [16] | Makarov VL, Makarov IL, Prikazčikov VG: Tochnye raznostnye shemy i shemy liubogo poriadka tochnosti dlia sistem obyknovennyh differencialnych uravnenij vtorogo poriadka.Differentsial’nye Uravneniya 1979, 15: 1194-1205. · Zbl 0422.34011 |
| [17] | Makarov VL, Samarskiĭ AA: Exact three-point difference schemes for second-order nonlinear ordinary differential equations and their realization.Doklady Akademii Nauk SSSR 1990,312(4):795-800. |
| [18] | Makarov VL, Samarskiĭ AA: Realization of exact three-point difference schemes for second-order ordinary differential equations with piecewise smooth coefficients.Doklady Akademii Nauk SSSR 1990,312(3):538-543. |
| [19] | Ronto M, Samoilenko AM: Numerical-Analytic Methods in the Theory of Boundary-Value Problems. World Scientific, New Jersey; 2000:x+455. · Zbl 0957.65074 · doi:10.1142/3962 |
| [20] | Samarskiĭ AA, Makarov VL: Realization of exact three-point difference schemes for second-order ordinary differential equations with piecewise-smooth coefficients.Differentsial’nye Uravneniya 1990,26(7):1254-1265, 1287. · Zbl 0711.65060 |
| [21] | Scott, MR; Watts, HA; Aziz, AK (ed.), A systematized collection of codes for solving two-point boundary value problems, 197-227 (1976), New York |
| [22] | Stoer J, Bulirsch R: Introduction to Numerical Analysis, Texts in Applied Mathematics. Volume 12. Springer, New York; 2002:xvi+744. · Zbl 1004.65001 · doi:10.1007/978-0-387-21738-3 |
| [23] | Tihonov AN, Samarskiĭ AA: Homogeneous difference schemes.Žurnal Vyčislitel’ noĭ Matematiki i Matematičeskoĭ Fiziki 1961, 1: 5-63. · Zbl 0131.34102 |
| [24] | Tihonov AN, Samarskiĭ AA: Homogeneous difference schemes of a high order of accuracy on non-uniform nets.Žurnal Vyčislitel’ noĭ Matematiki i Matematičeskoĭ Fiziki 1961, 1: 425-440. |
| [25] | Trenogin VA: Functional Analysis. “Nauka”, Moscow; 1980:496. · Zbl 0517.46001 |
| [26] | Troesch BA: A simple approach to a sensitive two-point boundary value problem.Journal of Computational Physics 1976,21(3):279-290. 10.1016/0021-9991(76)90025-5 · Zbl 0334.65063 · doi:10.1016/0021-9991(76)90025-5 |
| [27] | Wallisch W, Hermann M: Schießverfahren zur Lösung von Rand- und Eigenwertaufgaben, Teubner-Texte zur Mathematik. Volume 75. BSB B. G. Teubner, Leipzig; 1985:180. · Zbl 0577.65065 |
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.
