With how much confidence can we rely on computer simulations? For example, suppose we use a numerical method to approximate the general solution of a differential equation. Does the approximation exhibit the same qualitative behavior as the exact solution? In this talk, we shall show that the answer to this question is YES for several cases. In so doing, we will see how some techniques in dynamical systems can provide new insight into numerical algorithms. First, we shall describe the connection between dynamical systems and numerical methods. Next, some classical stability theorems in dynamical systems will be summarized. Third, we will present our results and explain the ideas of the proofs. At the end, we will present some questions for further study.