gives the
Jacobian matrix 
and 
denotes the update
vector.
For the Newton iteration method the higher order terms of (4.1-23)
are neglected and the linearized equation system (4.1-24) at the
iteration is solved instead.
is referred to as right-hand-side (RHS)
vector. To solve the nonlinear system we have to start the Newton iteration
with an initial guess 
for the solution variables. This initial
guess is coming from the initial conditions at the first time step or taken
from the previous time step. The new solution of the equation reads:
In the case of a linear equation system, 
would be the exact
solution. As diffusion systems are nonlinear the Newton method exhibits a
tendency to overestimate the update 
. This leads to
intermediate results which can lie outside the convergence region of the
Newton scheme or causes numerical problems which totally prevent
convergence. Hence a damping factor d;SPMlt;1 is introduced to avoid this
overshoot. Only a fraction of the update vector is added to the
iterate solution vector. Several successfully applied damping algorithms can
be found in the literature [Ban81] [Ban80] [Deu74].
The iteration scheme is terminated when the norm of the update vector 
is smaller than a specified Newton accuracy
.
Additionally, the residual of the nonlinear system is checked to be smaller
than a chosen absolute accuracy 
.
