Spurious solutions of numerical methods for initial value problems

Abstract

It is well known that some numerical methods for initial value problems admit spurious limit sets. Here the existence and behaviour of spurious solutions of Runge-Kutta, linear multistep and predictor-corrector methods are studied in the limit as the step-size h→0. In particular, it is shown that for ordinary differential equations defined by globally Lipschitz vector fields, spurious fixed points and period 2 solutions cannot exist for h arbitrarily small, whilst for locally Lipschitz vector fields, spurious solutions may exist for h arbitrarily small, but must become unbounded as h→0. The existence of spurious solutions is also studied for vector fields merely assumed to be continuous, and an example is given, showing that in this case spurious solutions may remain bounded as h→0