Stability and Convergence Analysis of Numerical Schemes for Parabolic PDEs Modelling Heat Diffusion

Abstract: We study stability and convergence properties of standard finite‐difference time-stepping schemes for the one-dimensional heat equation u_t=αu_xx on [0,1] with homogeneous Dirichlet boundaries. Using von Neumann analysis, we recover the classical results: FTCS is conditionally stable with a CFL restriction on r=α∆t/(∆x)^2 , while BTCS and Crank-Nicolson (CN) are unconditionally stable. Local truncation error analysis shows FTCS is first-order in time and second-order in space, and CN achieves second-order accuracy in both time and space. Numerical experiments with the analytical benchmark e^(-απ^2 t) sin⁡(πx) confirm the theory: log-log error plots exhibit slopes consistent with the predicted orders, and 3D surface/contour comparisons show close agreement between CN and the exact solution. A performance summary highlights the trade-offs between explicit simplicity versus time-step restrictions, implicit robustness versus linear-solve cost, and CN's balanced accuracy-stability-cost profile.

Keywords: Heat equation; parabolic PDE; finite differences; von Neumann stability; Crank-Nicolson; FTCS; BTCS; convergence; truncation error; error norms.