A new single-segment group explicit iteration scheme based on cubic spline approximation for 1D nonlinear parabolic partial differential equations

In this article, we present a new two-step cubic spline method of O(2,4) for estimating the numerical solution of a 1D nonlinear parabolic partial differential equation with appropriate conditions at the initial time level and Dirichlet conditions along the boundary. Furthermore, we present a new single-segment, group-explicit iteration procedure to solve a nonlinear system of difference equations at each subsequent time level. The convergence of this new iterative scheme is analysed. We have applied the proposed numerical procedures to solve benchmark problems, namely the Burgers-Huxley, Burgers-Fischer, and singular parabolic equations. The effectiveness of the proposed iteration schemes is validated by numerical results, which are compared with the corresponding two-step Newton-SoR and Newton-AGE iteration procedures.