TITLE:
An Efficient Explicit Scheme for Solving the 2D Heat Equation with Stability and Convergence Analysis
AUTHORS:
Md Nazmul Haque, Ruma Akter, Md Shahadat Hossain Mojumder
KEYWORDS:
Explicit Scheme, Finite Difference Method, CFL Condition, Stability, Convergence, Von-Neumann Method, Error Norms
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.7,
July
11,
2025
ABSTRACT: This paper presents a comprehensive numerical study of the two-dimensional time-dependent heat conduction equation using the Forward Time Centered Space (FTCS) finite difference scheme. The heat equation is a fundamental parabolic partial differential equation, models the diffusion of thermal energy in a medium and is applicable in areas such as thermal insulation design, microchip cooling, and biological heat transfer. Due to the limitations of analytical methods in handling complex geometries and boundary conditions, we employ the FTCS scheme. The problem is formulated with Dirichlet boundary conditions and a sinusoidal initial condition for which an exact analytical solution is known. We derive the FTCS discretization using Taylor series-based approximations and perform a detailed von Neumann stability analysis to establish the Courant-Friedrichs-Lewy (CFL) condition. The scheme’s performance is evaluated through numerical simulations on a uniform grid, with results compared against the exact solution. Simulation results show that the FTCS scheme achieves L2 and max-norm errors on the order of 10−11 and 10−10, respectively, under stable conditions. Graphical comparisons further demonstrate excellent agreement between numerical and analytical solutions. Overall, the FTCS method proves to be a robust and reliable tool for solving heat conduction problems, provided the stability criterion is satisfied.