Spurious solutions of numerical methods for initial value problems

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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. It is shown that if spurious fixed points or period 2 solutions of continuous problems exist for h arbitrarily small, then as h→0 spurious solutions either converge to steady solutions of the underlying differential equation or diverge to infinity. A necessary condition for the bifurcation spurious solutions from h=0 is derived. To prove the above results for implicit Runge-Kutta methods, an additional assumption on the iteration scheme used to solve the nonlinear equations defining the method is needed; an example of a Runge-Kutta method which generates a bounded spurious solution for a smooth problem with h arbitrarily small is given, showing that such an assumption is necessary. It is also shown that an Adams-Bashforth/Adams-Moulton predictor-corrector method in PCm implementation can generate spurious fixed point solutions for any m.

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