@code
...
T=0.14
- Number of active cells: 1801
- Number of degrees of freedom: 7676
+ Number of active cells: 1807
+ Number of degrees of freedom: 7696
NonLin Res Lin Iter Lin Res
_____________________________________
- 1.033e-02 0007 7.41e-13
- 4.079e-05 0007 3.24e-15
- 3.475e-09 0008 1.75e-20
- 5.441e-16 (converged)
+ 7.015e-03 0008 3.39e-13
+ 2.150e-05 0008 1.56e-15
+ 2.628e-09 0008 5.09e-20
+ 5.243e-16 (converged)
T=0.16
- Number of active cells: 1804
- Number of degrees of freedom: 7684
+ Number of active cells: 1807
+ Number of degrees of freedom: 7696
NonLin Res Lin Iter Lin Res
_____________________________________
- 1.116e-02 0007 9.05e-13
- 5.045e-05 0007 4.37e-15
- 4.986e-09 0008 3.80e-20
- 5.720e-16 (converged)
+ 7.145e-03 0008 3.80e-13
+ 2.548e-05 0008 7.20e-16
+ 4.063e-09 0008 2.49e-19
+ 5.970e-16 (converged)
T=0.18
Number of active cells: 1807
NonLin Res Lin Iter Lin Res
_____________________________________
- 1.241e-02 0007 2.82e-13
- 7.053e-05 0007 3.71e-15
- 9.275e-09 0007 3.89e-19
- 6.131e-16 (converged)
+ 7.395e-03 0008 6.69e-13
+ 2.867e-05 0008 1.33e-15
+ 4.091e-09 0008 3.35e-19
+ 5.617e-16 (converged)
...
@endcode
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. 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
-movie, the results looks like this:
-
-<img src="http://www.dealii.org/images/steps/developer/step-33.slide.gif" alt="">
-
-As we see, the heavy mass of fluid is drawn down the slope by gravity, where
-it collides with the ski lodge and is flung into the air! Hopefully everyone
-escapes!
-
-We can also visualize the evolution of the adaptively refined grid:
-
-<img src="http://www.dealii.org/images/steps/developer/step-33.slide_adapt.gif" alt="">
-
-The adaptivity follows and precedes the flow pattern, based on the heuristic
-refinement scheme discussed above.
-
-
-
-
-<a name="extensions"></a>
-<h3>Possibilities for extensions</h3>
-
-<h4>Adaptive time stepping</h4>
-
-In the computations shown above, we use a fixed time step. This would
-appear sub-optimal: we should take larger steps whenever the problem
-is still solvable with Newton's method in 3 or 4 iterations; certainly
-more importantly we should take smaller steps whenever we can't solve
-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:
+the very rapid convergence we can see here. This holds until
+times up to $t=1.9$ at which time the nonlinear iteration reports a
+lack of convergence:
@code
...
-T=8.3
- Number of active cells: 2314
- Number of degrees of freedom: 10264
+T=1.88
+ Number of active cells: 2119
+ Number of degrees of freedom: 9096
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)
+ 2.251e-01 0012 9.78e-12
+ 5.698e-03 0012 2.04e-13
+ 3.896e-05 0012 1.48e-15
+ 3.915e-09 0012 1.94e-19
+ 8.800e-16 (converged)
-T=8.32
- Number of active cells: 2320
- Number of degrees of freedom: 10292
+T=1.9
+ Number of active cells: 2140
+ Number of degrees of freedom: 9184
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
+ 2.320e-01 0013 3.94e-12
+ 1.235e-01 0016 6.62e-12
+ 8.494e-02 0016 6.05e-12
+ 1.199e+01 0026 5.72e-10
+ 1.198e+03 0002 1.20e+03
+ 7.030e+03 0001 nan
+ 7.030e+03 0001 nan
+ 7.030e+03 0001 nan
+ 7.030e+03 0001 nan
+ 7.030e+03 0001 nan
+ 7.030e+03 0001 nan
----------------------------------------------------
-Exception on processing:
+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:
+An error occurred in line <2476> of file <\step-33.cc> in function
+ void Step33::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:
+Additional Information:
No convergence in nonlinear solver
--------------------------------------------------------
----------------------------------------------------
@endcode
-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.
+We may find out the cause and possible remedies by looking at the animation of the solution.
+
+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
+movie, the results of last several time steps looks like this:
+
+<img src="file:///Users/qiaol/devel/dealii/examples/step-33/step-33.oscillaton.gif " alt="" height="300">
+
+As we see, when the heavy mass of fluid hits the left bottom corner,
+some oscillation occurs and lead to the divergence of the iteration. A lazy solution to
+this issue is add more viscosity. If we set the diffusion power $\eta = 1.5$ instead of $2.0$,
+the simulation would be able to survive this crisis. Then, the result looks like this:
+
+
+<img src="file:///Users/qiaol/devel/dealii/examples/step-33/step-33.slide.ed2.gif " alt="" height="300">
+
+The heavy mass of fluid is drawn down the slope by gravity, where
+it collides with the ski lodge and is flung into the air! Hopefully everyone
+escapes! And also, we can see the boundary between heavy mass and light mass blur quickly
+due to the artificial viscosity.
+
+We can also visualize the evolution of the adaptively refined grid:
+
+<img src="file:///Users/qiaol/devel/dealii/examples/step-33/step-33.slide.adapt.ed2.gif " alt="" height="300">
+
+The adaptivity follows and precedes the flow pattern, based on the heuristic
+refinement scheme discussed above.
-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.
+
+<a name="extensions"></a>
+<h3>Possibilities for extensions</h3>
+
<h4>Stabilization</h4>
The numerical scheme we have chosen is not particularly
-stable. Furthermore, it is known how to make it more stable, for
-example by using streamline diffusion or least-squares stabilization
-terms.
+stable when the artificial viscosity is samll while is too diffusive when
+the artificial viscosity is large. Furthermore, it is known there are more
+advanced techniques to stabilize the solution, for example streamline
+diffusion, least-squares stabilization terms, entropy viscosity.
+
<h4>Better linear solvers</h4>
projects / dealii.git / commitdiff