SSPH-Newton Iterative Method for Solving Nonlinear Equations

In this paper, a new kernel approximation is constructed based on the basic principle of kernel approximation of meshless symmetric smooth particle hydrodynamics (SSPH) method. The first derivative is calculated by constructing a symmetric matrix through Taylor series expansion. Replacing the derivative of Newton’s method, the SSPH-Newton iterative method for solving nonlinear equations is proposed. The advantage of this method is that it does not need to evaluate the derivative and overcomes the shortcomings of Newton’s method. The quadratic convergence of the new method is proved. Numerical and application examples show that the method has the same accuracy as Newton method, the efficiency index is higher than Newton method, and the calculation amount is lower than Newton method.