A New SPH Iterative Method for Solving Nonlinear Equations

In this paper, based on the basic principle of the SPH method's kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton's method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton's method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton's method.