Analysis of Stability Regions of Numeric Methods Using the Time Scale Calculus

Abstract

Working with the time scale calculus, a technique created to unify difference equations and differential equations, we establish a connection between the stability region in the complex plane for a given approximation method, and the region in the complex plane where the generalized exponential function converges to 0. We make this connection for explicit and implicit Euler methods, as well for the explicit Runge-Kutta second-order method. Doing this, we find a very natural relationship between the exponential function corresponding to implicit Euler and the exponential function corresponding to explicit Euler. We then investigate this relationship for explicit Runge-Kutta method.