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Abstract. This paper concerns the theoretical analysis of step-by-step meth-ods for solving initial value problems in ordinary and partial differential equa-tions.The main theorem of the paper answers a natural question arising in thelinear stability analysis of such methods. It guarantees a (strong) version ofnumerical stability—under a stepsize restriction related to the stability regionof the numerical method and to a circle condition on the differential equation.The theorem also settles an open question related to the properties total-variation-diminishing, strong-stability-preserving, monotonic and (total-variation-)bounded. Under a monotonicity condition on the forward Eulermethod, the theorem specifies a stepsize condition guaranteeing boundednessfor linear problems.The main theorem is illustrated by applying it to linear multistep methods.For important classes of these methods, conclusions are thus obtained whichsupplement earlier results in the literature.

Stability and boundedness in the numerical solution of initial value problems