Symmetric boundary value methods for second order initial and boundary value problems
This paper is concerned with the numerical solution of both initial and boundary value problems for special second order equations \(y^{\prime\prime}(t)= f(y(t))\) by means of the so called Boundary Value Methods proposed by \textit{L. Brugnano} and \textit{D. Trigiante} [Solving differential equations by multistep. Initial and boundary value methods. Stability and Control: Theory, Methods and Applications. 6. London: Gordon and Breach (1998; Zbl 0934.65074)]. Here only symmetric multistep methods \( \rho (E) y_{n+j}= h^2 \sigma (E) f_{n+j}\) have been considered and some results on the stability and conditioning of these methods derived from the study of the boundary loci \( D_1= \{ \rho (e^{i \theta} )/ \sigma (e^{i \theta}; \theta \in [0, \pi) \}\) and \( D_2= \{ \rho (e^{x} )/ \sigma (e^{x}; x \in [0, + \infty) \}\) have been obtained. Finally, some numerical experiments with the problems of Kepler and Troetsch are presented to show how BVM based on symmetric second order multistep methods can be used to deal with these problems.
scientific article; zbMATH DE number 5147137