absolute stability

Whether a numerical method's own error stays bounded (rather than growing without limit) when it's run on a simple decaying test equation.

A numerical method is absolutely stable at $z=h\lambda$ (for the linear test equation $y'=\lambda y$ , $\operatorname{Re}(\lambda)<0$ ) if the error it introduces does not grow from one step to the next at that $z$ . The set of every $z$ where this holds is the method's region of absolute stability -- a region in the complex $h\lambda$ -plane, not just an interval, since $\lambda$ may itself be complex. A method's region size is a SEPARATE property from its order: two methods that are both 4th-order accurate can have absolute-stability regions differing in size by an order of magnitude.

Examples

Euler's method's region of absolute stability is the disk $|z+1|\le1$ in the complex plane -- for $\lambda=-100$ , that already forces $h\le0.02$ regardless of how smooth the true solution is.