The Fixed Point Iteration and Newton's Methods for the Nonlinear Wave Equation

The paper deals with the numerical solution of the nonlinear wave equation. The problem is solved by combining two methods - the generalized a-method for time discretization and the finite element method for space discretization. The nonlinear system of algebraic equations resulted from the corresponding discretization is solved by the Newton method. Newton's method shows fast convergence and takes several iterations in each time step. The aim of the paper is to study how the use of the fixed point iteration method, instead of Newton's method, influences the performance of finding numerical solution. As it can be concluded from the obtained results, both methods are comparable in terms of iterations while the run time of Newton's method is higher since it requires calculating the jacobian in each time step.