# Numerical Stability

The definition of numerically stability is similar to the stability of mechanical systems. A numerical procedure is stable if small perturbations of initial data result in small changes in the numerical solution.

It is worthwhile to comment the difference between physical stability and numerical
stability. Numerical instabilities arise from the discretization of the governing equations
of the system, whereas physical instabilities are instabilities in the solutions of the
governing equations independent of the numerical discretization. Usually numerical stability
is only examined for physically stable cases. For this reason in the simulation of the
physically unstable processes, it is not guaranteed to track accurately the numerical
instabilities. Numerical stability of a physically unstable process cannot be examined by
the definition given above. You establish the numerical stability criteria on the physically
stable system and suppose that any stable algorithm for a stable system remains stable on an
unstable system. ^{1}

On the other hand, the numerical stability of time integrators discussed in the literature concerns generally linear systems and extrapolated to nonlinear cases by examining linearized models of nonlinear systems. The philosophy is: if a numerical method is unstable for a linear system, it will be certainly unstable for nonlinear systems as linear cases are subsets of the nonlinear cases. Therefore, the stability of numerical procedures for linear systems provides a useful guide to explore their behavior in a general nonlinear case.

To study the stability of the central difference time integration scheme, you establish the necessary conditions to ensure that the solution of equations is not amplified artificially during the step-by-step procedure. Stability also means that the errors due to round-off in the computer, do not grow in the integration. It is assured if the time step is small enough to accurately integrate the response in the highest frequency component.

^{1}Belytschko T., Wing Kam Liu, and Moran B., “Finite Elements for Nonlinear Continua and Structures”, John Wiley, 1999.