Spectral Optimized Multiderivative Hybrid Block Method for Fitzhugh–Nagumo Equations

Abstract

The Fitzhugh–Nagumo equation, a key model for excitable systems in biology and neuroscience, requires efficient numerical methods due to its nonlinear nature. A spectral optimized multiderivative hybrid block method is proposed, constructed using a multistep collocation and interpolation technique with an approximated power series as the basis function. Incorporating two optimal intra-step points, the method demonstrates improved accuracy, with its consistency, convergence, and absolute stability rigorously analyzed. By combining the optimized multiderivative hybrid block method in time with a spectral collocation method in space, the approach demonstrates potency and flexibility in solving partial differential equations. Prior to using the spectral method, the partial differential equation is linearized using a linear partition technique. Numerical experiments confirm the accuracy and efficiency of the method compared to existing methods, demonstrating the potential of the method as a robust framework for solving partial differential equations requiring both high accuracy and stability.