ABSTRACT
The subject of multistep stability is much more complicated than the corresponding Runge–Kutta (RK) topic. With RK methods their consistency guarantees convergent processes, but this is not the case in the multistep situation, in which rounding error at any step of the calculation must be damped out by subsequent steps. While single-step methods sometimes are required to be absolutely stable, they exhibit no instability when applied to differential equations with positive transients. However, multistep processes can contain characteristic values which destroy all numerical solutions. Zero or strong stability must be a property of any multistep process in any application. Other factors which are associated with non-convergence of multistep processes are generally placed under the umbrella of weak stability properties. As with RK formulae, absolute stability regions in the complex plane relating to multistep methods can be plotted. However, the investigation of relative stability needs a consideration of the separate roots.
