Stability of one-step and linear multistep methods - a matrix technique approach

Abstract

We investigate the stability of one-step and linear multistep methods from a new direction. Our aim is to modify the long and technical proof which is consequently omitted in almost every textbook and make it user-friendly. In the literature the techniques of numerical solution of initial value problems and boundary value problems seem to have almost nothing in common which is quite surprising. Our new approach uses matrix techniques opposed to the usual recursion approach, thus applying the techniques of boundary value problems to initial value problems. Even though the proof remains long, it is easier to follow and connects two seemingly separated areas, consequently this approach might have educational profit.

Keywords

• stability
• linear multistep methods
• m-matrix theory