BS2 methods for semi-linear second order boundary value problems. (English) Zbl 1338.65190

Summary: A new class of Linear Multistep Methods based on B-splines for the numerical solution of semi-linear second order Boundary Value Problems is introduced. The presented schemes are called BS2 methods, because they are connected to the BS (B-spline) methods previously introduced in the literature to deal with first order problems. We show that, when using an even number of steps, schemes with good general behavior are obtained. In particular, the absolute stability of the 2-step and 4-step BS2 methods is shown. Like BS methods, BS2 methods are of particular interest, because it is possible to associate with the discrete solution a spline extension which collocates the differential equation at the mesh points.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations


Full Text: DOI


[1] Amodio, P.; Brugnano, L., The conditioning of Toeplitz band matrices, Math. Comput. Model., 23, 29-42, (1996) · Zbl 0858.65045
[2] Amodio, P.; Sgura, I., High-order finite difference schemes for the solution of second-order BVPs, J. Comput. Appl. Math., 176, 59-76, (2005) · Zbl 1073.65061
[3] Ascher, U. M.; Mattheij, R. M.M.; Russel, R. D., Numerical solution of boundary value problems for ordinary differential equations, (1988), Prentice Hall Englewood Cliffs, New York
[4] Ascher, U. M.; Christiansen, J.; Russell, R. D., Collocation software for boundary-value odes, ACM Trans. Math. Softw., 7, 209-222, (1981) · Zbl 0455.65067
[5] Bader, G.; Ascher, U. M., A new basis implementation for a mixed order boundary value ODE solver, SIAM J. Sci. Stat. Comput., 8, 483-500, (1987) · Zbl 0633.65084
[6] Brugnano, L.; Trigiante, D., Solving differential problems by multistep initial and boundary value methods, (1998), Gordon and Breach Science Publishers Amsterdam · Zbl 0934.65074
[7] de Boor, C., A practical guide to splines, (2001), Springer-Verlag New York · Zbl 0987.65015
[8] Mazzia, F.; Cash, J. R.; Soetaert, K., Solving boundary value problems in the open source software R: package bvpsolve, Opuscula Math., 34, 387-403, (2014) · Zbl 1293.65104
[9] Mazzia, F.; Sestini, A., The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions, BIT Numer. Math., 49, 611-628, (2009) · Zbl 1181.65017
[10] Mazzia, F.; Sestini, A.; Trigiante, D., B-spline multistep methods and their continuous extensions, SIAM J. Numer. Anal., 44, 1954-1973, (2006) · Zbl 1128.65057
[11] Mazzia, F.; Sestini, A.; Trigiante, D., BS linear multistep methods on non-uniform meshes, J. Numer. Anal. Ind. Appl. Math., 1, 131-144, (2006) · Zbl 1120.65090
[12] Mazzia, F.; Sestini, A.; Trigiante, D., The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes, Appl. Numer. Math., 59, 723-738, (2009) · Zbl 1161.65057
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.