Analysis and Implementation of Numerical Solutions for Partial Differential Equations

Abstract

This study explores the application of the finite difference method for solving one-dimensional wave and heat equations, extending from basic boundary value problems to more complex time-dependent scenarios. By discretizing both spatial and temporal domains, we developed a numerical scheme for approximating solutions to these partial differential equations effectively. Our analysis underscores the importance of numerical stability, specifically the role of the Courant–Friedrichs–Lewy (CFL) condition in ensuring accurate results. We found that improper balance between spatial step size h and temporal step size k, particularly when h/k is below a critical threshold related to the sound speed c, can lead to instability, manifesting as oscillations and overflow errors. Despite these challenges, the finite difference method remains a powerful tool for solving time-dependent PDEs when stability conditions are properly managed. Practical examples and visualizations in the Jupyter notebook highlight these concepts and their implications.