Split Newton iterative algorithm and its application

Inspired by some implicit–explicit linear multistep schemes and additive Runge–Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction–diffusion equations, such as Burger’s–Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.