The single degree of freedom equation for time exact and time discrete problems is analyzed here. The equations are written with the initial conditions. Also, using the time exact equation the time discrete equation can be derived. This is possible because the process of time discretization and modal decomposition are linear ones.
In this lesson the stability of the equations will be discussed. First the stability of time exact systems is discussed. The use of the modal index is dropped. An exact solution for the ODE is written which satisfies the initial conditions as well. This study helps in understanding the exact behavior of the algorithmic equations. After the time-exact equations the time discrete equations are discussed. Amplification factor, stability, and spatial discretization ae discussed for these equations.