Difference schemes for nonlinear BVPs using Runge-Kutta IVP-solvers
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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 ᏻ(|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 the 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. Copyright © 2006 I. P. Gavrilyuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction This paper deals with boundary value problems (BVPs) of the form u (x) + A(x)u = f(x,u),
x ∈ (0,1),
B0 u(0) + B1 u(1) = d,
(1.1)
where A(x),B0 ,B1 , ∈ Rd×d ,
rank B0 ,B1 = d,
f(x,u),d,u(x) ∈ Rd ,
(1.2)
and u is an unknown d-dimensional vector-function. On an arbitrary closed irregular grid
h = x j : 0 = x0 < x1 < x2 < · · · < xN = 1 , ω
Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 12167, Pages 1–29 DOI 10.1155/ADE/2006/12167
(1.3)
2
Difference schemes for BVPs
there exists a unique two-point exact difference scheme (EDS) such that its solution co h . Algorithmical incides with a projection of the exact solution of the BVP onto the grid ω realizations of the EDS are the so-called truncated difference schemes (TDSs). In [14] an algorithm was proposed by which for a given integer m an associated TDS of the order of accuracy m (or shortly m-TDS) can be developed. The EDS and the corresponding three-point difference schemes of arbitrary order of accuracy m (so-called truncated difference schemes of rank m or shortly m-TDS) for BVPs for systems of second-order ordinary differential equations (ODEs) with piecewise continuous coefficients were constructed in [8–18, 20, 23, 24]. These ideas were further developed in [14] where two-point EDS and TDS of an arbitrary given order of accuracy for problem (1.1) were proposed. One of the essential parts of the resulting algorithm was the computation of the fundamental matrix which influenced considerably its complexity. Another essential part was the use of a Cauchy problem solver (IVP-solver) on each subinterval [x j −1 ,x j ] where a one-step Taylor series method of the order m has been chosen. This supposes the calculation of derivatives of the right-hand side which negatively influences the efficiency of the algorithm. The aim of this paper is to remove these two drawbacks and, therefore, to improve the computational complexity and the effectiveness of TDS for problem (1.1). We propose a new implementation of TDS with the following main features: (1) the
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