Damped Newton method

One method of improving the convergence properties of Newton s method is to implement a damping strategy [95]. In a damped Newton method, Eq. 15.46 becomes... 

The solution method, which uses a combination of a damped Newton method and time marching, is essentially the same as that introduced in Section 15.5 [159]. The differences have to do with the structure of the Jacobian matrix and the need for mesh adaptation. 

This appendix explains how to use DDAPLUS to solve nonlinear initial-value problems containing ordinary differential equations with or without algebraic equations, or to solve purely algebraic nonlinear equation systems by a damped Newton method. Three detailed examples are given. 

Iwork(16) contains the maximum number of iterations permitted for the damped Newton method in an integration step. The default value of 4 is used if the input value is anything less. 

The time step, At, is used to switch the method from being a relaxation method to a global Newton method. When the time step is small, e.g., if = 0.1, then the changes in the independent variables are small. The method performs like a damped Newton-Raphson method, where the steps are small but in the direction of the solution and without any oscillation. When the value of At is large, i.e., At = 1000, the method performs like a Newton-Raphson method. The value of At at each column trial determines the speed and stability of the method, The units of the time step are the same as the flows to and from the column. The calculation sequence of the Ketchum method is as follows ... 

Fig. 15.6 Graphical illustration of the damping strategy for Newton s method.

One method, which avoids the problem with undesired negative eigenvalues of the Hessian, and which introduces an automatic damping of the rotations, is the augmented Hessian method (AM). To describe the properties of this method, let us again consider the Newton-Raphson equation (4 4) ... 

The Newton-Raphson technique is also modified in this method. A damping factor a, between zero and one, is applied to the corrections as above. The way a is calculated, however, is different from the Naphtali-Sandholm method. 

---