Explicit vs. Implicit ODE Solvers: Stability, Robustness, and Practical Implications

This article delves into the strengths and weaknesses of explicit and implicit ordinary differential equation (ODE) solvers. While implicit methods are often considered more robust due to their superior stability, the author argues that explicit methods can be preferable for certain problems, especially those requiring the preservation of oscillations. Through linear ODE analysis, the concept of stability regions, and real-world examples (like cooling and oscillatory systems), the article illustrates the performance of both methods in different scenarios. It emphasizes that selecting the appropriate solver requires a nuanced understanding of the problem at hand, rather than a blanket approach.