How long do numerical chaotic solutions remain valid

Dynamical conditions for the loss of validity of numerical chaotic solutions of physical systems are already understood. However, the fundamental questions of {open_quotes}how good{close_quotes} and {open_quotes}for how long{close_quotes} the solutions are valid remained unanswered. This work answers these questions by establishing scaling laws for the shadowing distance and for the shadowing time in terms of physically meaningful quantities that are easily computable in practice. The scaling theory is verified against a physical model. {copyright} {ital 1997} {ital The American Physical Society}