Stability and boundedness in the numerical solution of initial value problems

This paper concerns the theoretical analysis of step-by-step methods for solving initial value problems in ordinary and partial differential equations. The main theorem of the paper answers a natural question arising in the linear stability analysis of such methods. It guarantees a (strong) version of numerical stability – under a stepsize restriction related to the stability region of the numerical method and to a circle condition on the differential equation. The theorem settles also an open question related to the properties total-variation-diminishing, strongstability-preserving, monotonic and (total-variation-)bounded. Under a monotonicity condition on the forward Euler method, the theorem specifies a stepsize condition guaranteeing boundedness for linear problems. The main theorem is illustrated by applying it to linear multistep methods. For important classes of these methods, conclusions are thus obtained which supplement earlier results in the literature. AMS subject classifications. 65L05, 65L06, 65L20, 65M12, 65M20.