stepping. Note that our implementation of the Newton iteration indeed shows
the expected quadratic convergence order: the norm of the nonlinear residual
in each step is roughly the norm of the previous step squared. This leads to
-the very rapid convergence we can see here.
+the very rapid convergence we can see here. This holds at least for
+times up to $t=8.32$ at which time the nonlinear iteration reports a
+lack of convergence; the cause and possible remedies are discussed below.
The result of running these computations is a bunch of output files that we
can pass to our visualization program of choice. When we collate them into a
the problem at hand any more with Newton's method. The latter is what
happens to the calculation the screen output of which was shown above:
@code
+...
+
+T=8.3
+ Number of active cells: 2314
+ Number of degrees of freedom: 10264
+
+ NonLin Res Lin Iter Lin Res
+ _____________________________________
+ 4.069e-02 0018 3.74e-12
+ 3.570e-04 0020 6.83e-15
+ 8.349e-07 0020 2.51e-17
+ 2.858e-11 (converged)
+
+T=8.32
+ Number of active cells: 2320
+ Number of degrees of freedom: 10292
+
+ NonLin Res Lin Iter Lin Res
+ _____________________________________
+ 5.607e-02 0240 4.36e-12
+ 5.807e-04 0300 1.54e-05
+ 1.538e-05 0300 1.22e-05
+ 1.218e-05 0300 9.15e-06
+ 9.146e-06 0300 4.27e-06
+ 4.264e-06 0300 3.69e-06
+ 3.693e-06 0300 3.31e-06
+ 3.308e-06 0300 2.81e-06
+ 2.809e-06 0300 2.77e-07
+ 2.781e-07 0300 2.33e-07
+ 2.328e-07 0300 1.89e-07
+
+
+----------------------------------------------------
+Exception on processing:
+--------------------------------------------------------
+An error occurred in line <3119> of file <step-33.cc> in function
+ void ConservationLaw<dim>::run() [with int dim = 2]
+The violated condition was:
+ nonlin_iter <= 10
+The name and call sequence of the exception was:
+ ExcMessage ("No convergence in nonlinear solver")
+Additional Information:
+No convergence in nonlinear solver
+--------------------------------------------------------
+
+Aborting!
+----------------------------------------------------
@endcode
-If this happens, it would be nice if we could either (i) detect the
-problem up front and reduce the time step before we even start the
-time step, or (ii) accept the failure at this time step and then
-simply start over from the previous time step trying with a reduced
-step size.
+
+From looking at the graphical output, it isn't immediately clear if
+there is a physical event that triggers this breakdown. However,
+whatever the matter, the solver should certainly not just break down.
+
+If this happens nevertheless, it would be nice if we could either (i)
+detect the problem up front and reduce the time step before we even
+start the time step, or (ii) accept the failure at this time step and
+then simply start over from the previous time step trying with a
+reduced step size.
<h4>Stabilization</h4>