Preserving Oscillatory Accuracy in Numerical Time-Stepping

This chapter helps readers build practical intuition for selecting reliable time integration techniques in computational work. It offers a hands-on introduction to how different time-stepping methods perform when solving oscillatory ordinary differential equations. It breaks down how numerical schemes influence amplitude and phase, showing how they can distort signals and what that means for accuracy and stability. Through a side-by-side look at explicit and implicit Euler methods, it highlights the balance between ease of use and robustness. Readers are also given simple tools to estimate errors without needing to run full-scale simulations. Using Fourier-based analysis, the chapter demystifies the origins of numerical dispersion and dissipation, offering insight into the behavior of more advanced methods like Runge-Kutta and multistep schemes.