Consistency, Stability, and Convergence Analysis of a Finite Difference–Based Method of Lines with Five-Point Discretization

This paper presents a detailed investigation into the consistency, stability, and convergence properties of the finite difference–based Method of Lines (FD-MoL) employing a five-point discretization for the numerical solution of parabolic partial differential equations. The heat equation without a source term is adopted as a benchmark problem to illustrate the methodology and theoretical analyses. In the FD-MoL framework, spatial derivatives are discretized using a high-order central finite difference scheme, while the temporal domain remains continuous, allowing for the application of robust ordinary differential equation (ODE) solvers.The consistency of the proposed scheme is established through local truncation error analysis, revealing first-order accuracy in time and secondorder accuracy in space. Numerical stability is examined using Von Neumann Fourier analysis, with results showing that the implicit formulation is unconditionally stable for all wave numbers and time-step sizes. Convergence is verified both analytically and in accordance with the Lax equivalence theorem, confirming that the numerical solution converges to the exact solution as the discretization parameters approach zero.The study highlights the effectiveness of the five-point stencil in enhancing spatial resolution while maintaining numerical robustness. These properties make the scheme highly suitable for large-scale and longtime simulations in science and engineering domains where high spatial accuracy and unconditional stability are essential.