Iterative solution of nonlinear difference equations for shock problems

The use of Newton's method for iterative solution of nonlinear difference equations for flows with shocks are described. It is assumed that conditions defining uniqueness of the solutions are satisfied. Newton's method is demonstrated to display a locally quadratic convergence but only a small domain of attraction, thereby requiring accurate initial vectors. Techniques for obtaining the needed vectors are defined: uses of globally convergent methods such as the parallel chord method, prior solution of smaller systems with a larger step-size and then interpolating to a finer grid or solution of the shock equation with a large disturbance first to increase the domain of attraction.