High order second derivative multistep collocation methods for ordinary differential equations

Document Type : Research Article


Marand Technical Faculty, University of Tabriz, Tabriz, Iran.


In this paper, we introduce second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.


Main Subjects

[1] Abdi, A. Construction of high order quadratically stable second derivative general linear methods for the numerical integration of stiff ODEs, J. Comput. Appl. Math. 303 (2016), 218–228.
[2] Abdi, A. and Hojjati, G. Implementation of Nordsieck second derivative methods for stiff ODEs, Appl. Numer. Math. 94 (2015), 241–253.
[3] Brunner, H. Collocation methods for Volterra integral and related func-tional equations, Cambridge University Press, 2004.
[4] Butcher, J.C. and Hojjati, G. Second derivative methods with RK sta-bility, Numer. Algorithms. 40 (2005), 415–429.
[5] Cash, J.R. Second derivative extended backward differentiation formulas for the numerical integration of stiff systems, SIAM J. Numer. Anal. 18 (1981), 21–36.
[6] D’Ambrosio, R., Ferro, M., Jackiewicz, Z. and Paternoster, B. Two-step almost collocation methods for ordinary differential equations, Numer. Algorithms. 53 (2010), 195–217.
[7] Enright, W.H. Second derivative multistep methods for stiff ordinary differential equations, SIAM. J. Numer. Anal. 7 (1974), 321–331.
[8] Fazeli, S. and Hojjati, G. Second derivative two-step collocation methods for ordinary differential equations, Appl. Numer. Math. 156 (2020), 514–527.
[9] Ferguson, D. Some interpol ation theorems for polynomials, J. Approx. Theory. 9 (1973), 327–348.
[10] Hairer, E.: https://www.unige.ch/~hairer/testset/testset.html.
[11] Hairer, E., Norsett, S.P. and Wanner, G. Solving ordinary differential equations I: Nonstiff problems, Springer-Verlag. second revised edition, 1993.
[12] Hairer, C.L.E. and Wanner, G. Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Sec-ond Edition. Springer Series in Computational Mathematics 31, Springer-Verlag, 2006.
[13] Hairer, E. and Wanner, G. Solving Ordinary Differential Equations II: Stiff and Differential´┐┐Algebraic Problems, Springer, Berlin, 2010.
[14] Lie, I. and Nørsett, S.P. Superconvergence for multistep collocation, Math. Comp. 52 (1989), 65–79.
[15] Liniger, W. and Willoughby, R.A. Efficient numerical integration of stiff systems of ordinary differential equations, Technical Report RC-1970, Thomas J. Watson Research Center, Yorktown Heihts, New York, 1976.
[16] Mazzia, F. and Magherini, C. Test set for initial value problem solvers, University of Bari. Italy, 2006.
[17] McLachlan, R.I. and Quispel, G.R.W. Geometric integrators for ODEs, J. Phys. A: Math. Gen. 39 (2006), 5251–5285.
[18] Schoenberg, I.J. On Hermite-Birkhoff interpolation, J. Math. Anal. Appl. 16 (1966), 538–543.