Long - Time Error Estimation and a Stability Indicator

Long-time error estimates are abstractly given for a large class of initial value problems without using the traditional concept of \numerical stability". Instead of numerical error propagation, we consider exact error propagation by splitting the error of a numerical initial value problem into local error and propagated global error in a way diierent from the traditional one. The advantage is that we can use whatever contraction property the differential equation has. The tradeoo is that we need to estimate the error of approximation of a nearby solution with a numerical initial value. Rigorous analysis of the global error bound is given for both non-stii and linear stii cases. For nonlinear stii problems, we propose a smoothing assumption for the problem, and a stability indicator for the numerical solution. If the smoothing assumption is valid and the indicator remains bounded during the computation , we have uniform error bounds on t 0 ; 1). The class of problems covered by this theory is characterized by the smoothing assumption. This approach of analysis separates the concept of the stability of a discrete process from that of the error propagation of numerical schemes and, therefore, makes the error analysis for complex equations treated by complicated schemes signiicantly easier. 1. Introduction Long-time error estimation for initial value problems for ODE's and PDE's is important for both the theory of numerical analysis and the practice of scientiic computation. The traditional error propagation analyses were based on the numerical schemes-numerical error propagation. For such analyses, the theory, which is often refered as numerical stability, was developed during the past years. But for complex nonlinear systems treated by a combination of numerical techniques, such as linearization, partially implicit schemes, local time-stepping, insuucient iterations for a local time step, etc., it is too diicult and tedious, if even possible, to carry out error propagation analysis for the numerical methods. In fact, the word \stability" is used in many cases where what is meant is actually \numerical error propagation". For linear equations, because of the additivity, stability of solution and boundedness of error propagation are equivalent. Unfortunately , additivity does not generalize to nonlinear equations. Consequently, generalization of the linear stability theory to nonlinear cases has met some hard obstacles, which leads us to the proposition that the theory of numerical stability