A Stability-Admissible Optimization Framework for Time-Step Selection in Finite Difference Heat Equation Solvers

Abstract

The one-dimensional heat equation is a fundamental parabolic partial differential equation used to model diffusive heat transfer processes. Although classical finite difference schemes are well established, their numerical accuracy depends on the selected time-step size. In practice, the time step is often chosen using stability restrictions such as the Courant--Friedrichs--Lewy (CFL) condition. However, stability does not necessarily correspond to minimum numerical error. Therefore, this study formulates the time-step selection problem as a constrained optimization problem within the stability-admissible region. The objective function is defined using the discrete maximum error between the numerical and exact solutions at the final simulation time. First, the time-step selection problem is formulated as a constrained optimization problem under the stability condition. Second, the numerical error is evaluated through the maximum norm of the difference between the numerical and exact solutions. Third, Particle Swarm Optimization (PSO) is used to determine suitable time-step values automatically. Since the optimization problem is bounded and nonlinear, PSO provides a simple derivative-free search strategy without requiring gradient information. Numerical results for the FTCS and Crank–Nicolson schemes show that different time-step values produce noticeably different numerical errors. Then, the optimization procedure consistently identifies time-step values with smaller error than conventional manually selected parameters. The results suggest that satisfying stability conditions alone is insufficient to obtain accurate numerical solutions.