Difference schemes for nonlinear BVPs on the half-axis

by    I. P. Gavriljuk, M. Hermann, M. V. Kutniv, V. L. Makarov

Preprint series: 06-25, Reports on Numerical Mathematics

I. P. Gavriljuk, M. Hermann, M. V. Kutniv, V. L. Makarov

65L10 Boundary value problems
65L12 Finite difference methods
65L20 Stability and convergence of numerical methods
65L50 Mesh generation and refinement
65L70 Error bounds
34B15 Nonlinear boundary value problems

Abstract: A scalar boundary value problem (BVP)
for a second order differential equation on the infinite interval [0,1) is considered.
Under some natural assumptions it is shown that on an arbitrary finite grid there
exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme
of which the solution coincides with the projection of the exact solution of the given
differential equation onto the underlying grid. A constructive method is proposed to
derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where
n is a freely selectable natural number and [·] denotes the entire part of the expression
in brackets. The n-TDS has the order of accuracy Żn = 2[(n+1)/2], i.e., the global error
is of the form O(|h|Żn), where |h| is the maximum step size. The n-TDS is represented
by a system of nonlinear algebraic equations for the approximate values of the exact
solution on the grid. Iterative methods for its numerical solution are discussed. The
theoretical and practical results are used to develop a new algorithm which has all the
advantages known from the modern IVP-solvers. Numerical examples are given which
illustrate the theorems presented in the paper and demonstrate the reliability of the
new algorithm.

Keywords: systems of nonlinear ordinary differential equations, difference scheme, exact difference scheme, truncated difference scheme of an arbitrary given accuracy order

Upload: 2006-12-22

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