Comparative Analysis of Finite Difference Schemes for the Two-Dimensional Heat Equation with Emphasis on the ADI Method

Abstract

The heat equation, a partial differential equation in Mathematics and Physics, plays a key role in various fields including probability theory, Brownian motion, and financial mathematics. This work presents a comparative study of various finite difference schemes, comprised of Forward Time Central Space (FTCS), Backward Time Central Space (BTCS), Crank-Nicolson (CN), and the Alternating Direction Implicit (ADI) method for approximating the two-dimensional Heat equation. The FTCS method, despite its simplicity and computational efficiency, is known to be conditionally stable and has limitations for large-scale problems. Implicit methods such as BTCS and CN offer improved stability at the cost of increased computational complexity. The ADI method, designed to handle large sparse systems efficiently, emerges as a robust alternative for high-dimensional and computationally intensive problems. This work employs these schemes, evaluates their stability and computational performance, and compares the simulation results against existing benchmarks. The findings highlight the advantages and limitations of each approach, with a particular focus on the suitability of the ADI method for practical applications involving two-dimensional heat conduction.