A New 3-Phase Cubic Spline Group Explicit Iteration Procedure for 1D Nonlinear Parabolic Partial Differential Equation: Application to Burgers–Huxley Equation

In this paper, we discuss a new 2-level implicit cubic spline method of order (2,4) for the numerical solution of 1D nonlinear parabolic equation [Formula: see text], [Formula: see text], [Formula: see text] subject to appropriate initial and Dirichlet boundary conditions prescribed on the boundary. Further, we discuss a new three-phase group explicit iteration procedure to solve nonlinear system of difference equations at each advanced time level as a numerical algorithm. The suggested three-phase iteration procedure is predicted to save the CPU-time as the common term is evaluated first at phase-I. The convergence analysis of the suggested iteration procedure is deliberated briefly. In application, we have solved a few benchmark problems including Burgers–Huxley, Burgers’ and singular parabolic equations. A comparison of the numerical results is carried out with the corresponding two-step Newton-AGE and Newton-SOR iteration procedures to demonstrate the usefulness of the purported iteration procedure in terms of CPU time and order of convergence.