Mathematical Formulation of a Symmetry-Compact Three-Step Algorithm for Computing the Spatio-Temporal Generalized FitzHugh–Nagumo Equations

This study presents the mathematical formulation of a symmetry-compact three-step algorithm (TSA) for the numerical computation of the spatio-temporal generalized FitzHugh–Nagumo equation (FHNE), a class of one-dimensional time-dependent initial-boundary value partial differential equations. The proposed symmetry-compact TSA is constructed using the Lagrange polynomial as the basis function, yielding a structurally balanced and computationally compact formulation with an inherent symmetry that facilitates automatic step-size adaptation over the integration interval. The symmetry-compact nature of the formulation enhances numerical stability while maintaining a reduced computational footprint, thereby improving both accuracy and efficiency when compared with existing numerical schemes. Prior to the application of the TSA, the FHNE is discretized in space, resulting in a system of ordinary differential equations suitable for time integration. Rigorous analyses of the stability and convergence properties of the symmetry-compact TSA are carried out to establish the reliability and robustness of the method. The performance of the proposed algorithm is quantitatively assessed using absolute error, maximum error, root mean square error, and central processing unit time for selected spatio-temporal test cases of the FHNE. The numerical results and corresponding solution profiles clearly demonstrate that the symmetry-compact TSA delivers superior accuracy, enhanced computational efficiency, and improved stability characteristics relative to existing methods, particularly in the presence of stiffness and chaotic dynamics.