The implicit Euler method for the numerical solution of singular initial value problems

• Published in: Applied numerical mathematics 2000-07, Vol.34 (2), p.231-252
• Main Authors: Koch, Othmar, Kofler, Peter, Weinmüller, Ewa B.
• Format: Article
• Language: English
• Subjects: Asymptotic error expansion for the global error of the Euler solution; Exact sciences and technology; Existence and uniqueness theory; Initial value problems; Iterated Defect Correction for singular initial value problems; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use; Singularity of the first kind; Stability and convergence of the implicit Euler method
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/S0168-9274(99)00130-0

The solvability of a certain class of singular nonlinear initial value problems is discussed. Particular attention is paid to the structure of initial conditions necessary for a bounded solution to exist. The implicit Euler rule applied to approximate the solution of the singular system is shown to be stable and to retain its classical convergence order. Moreover, the asymptotic error expansion for the global error of the above approximation is proven to have the classical structure. Finally, experimental results showing the feasibility of the approximation obtained by the Euler method to serve as a basic method for the acceleration technique known as the Iterated Defect Correction are presented.