Difference schemes for nonlinear BVPs on the half-axis

Abstract

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 accuracyn = 2[(n+1)/2], i.e., the global error is of the form O(|h|n), 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. , Vol. ? (200?), No. ?, 2000 Mathematics Subject Classification: 65L10, 65L12, 65L20, 65L50, 65L70, 34B15. COMPUTATIONAL METHODS IN APPLIED MATHEMATICSKeywords: systems of nonlinear ordinary differential equations, difference scheme, exact difference scheme, truncated difference scheme of an arbitrary given accuracy order.

Adaptive algorithms based on exact difference schemes for nonlinear BVPs on the half-axis

“…It has been shown in [4] that there exists a constant N 0 > 0 such that for all N N 0 the n-TDS (2.15)-(2.18), (2.11) possesses the order of accuracyn and the following estimate holds…”

“…For problem (1.1), there exists an EDS which can be derived in the following way (see [4] for details). First, the ordinary differential equation (1.1) is "doubly integrated" in a proper way over a small interval around a grid node x j (with a length proportional to the mesh size).…”

“…These proofs are given in our technical report [4] and in [5]. The corresponding code as well as various test examples are available under www.minet.uni-jena.de/~hermann/software.…”

“…For a variety of highly efficient numerical algorithms the class of EDS methods is the basis (see e.g. [4,5] and references therein for details and for comparison with other approaches).…”

“…Along with the description of the EDS we discuss its algorithmic realization by a special truncated difference scheme (TDS), a so-called n-TDS of rank n (see e.g. [4,5]). For each node x j , j = 1, 2, .…”