In the following section it is assumed that the thermal capacitance
depends on the temperature of the respective material. In analogy a nonlinear dependence of the thermal conductance
with respect to the temperature can be specified. However, for the sake of simplicity the heat conductance
is assumed to be constant in the following example.

(4.18) 
A finite volume discretization yields for the same fourpoint geometry as in the linear example of the last section:
By inserting the neighborhood information and setting the thermal generation term
and
, the following expression is obtained.
When using the Newton method it is always assumed that a given (already good) result is used to obtain a better result (see Fig. ). For the linear problem, a Newton scheme leads to a solution within one single step, because all higherorder terms vanish. For this reason in the linear case the choice of an initial guess does not play a role.
Figure:
The solutions of the linearized problems converge to the solution of the nonlinear problem using the Newton method.

For the nonlinear case, it is however important to chose a proper initial guess. In the following, the initial values
is chosen. Therefore, the expressions
and
yield



(4.20) 
First, the application of the
is performed as follows



(4.21) 
Repeated application of this step (eventually) leads to a solution of this equation system. However, it has to be considered that nonlinear methods have different rates of convergence and sometimes solutions can not be found at all [82]. Nevertheless, the presented method with linearized expressions can be used for the formulation of all kinds of nonlinear problems, the solution of these problems can be treated separately from the assembly.
Michael
20080116