Accuracy, Stability, and Convergence of Finite-Difference Schemes for the 1D Heat Equation

Abstract

This independent Computational Mathematics and Scientific Computing project investigates the accuracy, stability, and convergence of finite-difference methods for the one-dimensional heat equation. The project implements the Forward Time Centered Space (FTCS) and Crank-Nicolson numerical schemes in Python. It includes mathematical formulation, analytical reference solutions, stability analysis, convergence studies, error measurement, reproducible numerical experiments, and automated unit testing. NumPy and SciPy are used for numerical computation and banded linear system solving. The implementation includes discriminative tests for stable and unstable FTCS configurations and verifies approximately second-order convergence under the selected refinement strategies. This work was developed as an independent scientific coding portfolio project and has not undergone peer review.