Circumventing Ill-Conditioning Arising from Using Linear Multistep Methods in Approximating the Solution of Initial Value Problems

Abstract

When finding numerical solutions to stiff and nonstiff initial value problems using linear multistep methods, ill-conditioned systems are often encountered. In this paper, we demonstrate how this ill-conditioning can be circumvented without iterative refinement or preconditioning, by carefully choosing the grid point used in deriving the discrete scheme from the continuous formulation. Results of numerical experiments show that the new scheme perform very well when compared with the exact solution and results from an earlier scheme.

Keywords

• ill-conditioning
• linear multistep methods
• order
• (non)singularity
• underdetermined