+<a name="Results"></a>
+<h1>Results</h1>
+
+Running the program with default parameters in release mode takes about 1
+minute on a 4 core machine (with hyperthreading):
+@verbatim
+$ mpirun -np 4 ./step-69 | tee output
+Reading parameters and allocating objects... done
+
+ ####################################################
+ ######### #########
+ ######### create triangulation #########
+ ######### #########
+ ####################################################
+
+Number of active cells: 36864
+
+ ####################################################
+ ######### #########
+ ######### compute offline data #########
+ ######### #########
+ ####################################################
+
+Number of degrees of freedom: 37376
+
+ ####################################################
+ ######### #########
+ ######### set up time step #########
+ ######### #########
+ ####################################################
+
+ ####################################################
+ ######### #########
+ ######### interpolate initial values #########
+ ######### #########
+ ######### #########
+ ####################################################
+
+TimeLoop<dim>::interpolate_initial_values(t = 0)
+TimeLoop<dim>::output(t = 0, checkpoint = 0)
+
+ ####################################################
+ ######### #########
+ ######### enter main loop #########
+ ######### #########
+ ######### #########
+ ####################################################
+
+ ####################################################
+ ######### #########
+ ######### Cycle 000001 (0.0%) #########
+ ######### at time t = 0.00000000 #########
+ ######### #########
+ ####################################################
+
+[...]
+
+ ####################################################
+ ######### #########
+ ######### Cycle 007553 (100.0%) #########
+ ######### at time t = 3.99984036 #########
+ ######### #########
+ ####################################################
+
+TimeLoop<dim>::output(t = 4.00038, checkpoint = 1)
+
++------------------------------------------------------------------------+------------+------------+
+| Total CPU time elapsed since start | 357s | |
+| | | |
+| Section | no. calls | CPU time | % of total |
++------------------------------------------------------------+-----------+------------+------------+
+| discretization - setup | 1 | 0.113s | 0% |
+| offline_data - assemble lumped mass matrix, and c_ij | 1 | 0.167s | 0% |
+| offline_data - compute |c_ij|, and n_ij | 1 | 0.00255s | 0% |
+| offline_data - create sparsity pattern and set up matrices | 1 | 0.0224s | 0% |
+| offline_data - distribute dofs | 1 | 0.0617s | 0% |
+| offline_data - fix slip boundary c_ij | 1 | 0.0329s | 0% |
+| schlieren_postprocessor - compute schlieren plot | 201 | 0.811s | 0.23% |
+| schlieren_postprocessor - prepare scratch space | 1 | 7.6e-05s | 0% |
+| time_loop - setup scratch space | 1 | 0.127s | 0% |
+| time_loop - stalled output | 200 | 0.000685s | 0% |
+| time_step - 1 compute d_ij | 7553 | 240s | 67% |
+| time_step - 2 compute d_ii, and tau_max | 7553 | 11.5s | 3.2% |
+| time_step - 3 perform update | 7553 | 101s | 28% |
+| time_step - 4 fix boundary states | 7553 | 0.724s | 0.2% |
+| time_step - prepare scratch space | 1 | 0.00245s | 0% |
++------------------------------------------------------------+-----------+------------+------------+
+@endverbatim
+
+One thing that becomes evident is the fact that the program spends two
+thirds of the execution time computing the graph viscosity d_ij and about a
+third of the execution time in performing the update, where computing the
+flux $f(U)$ is the expensive operation. The preset default resolution is
+about 37k gridpoints, which amounts to about 148k spatial degrees of
+freedom in 2D. An animated schlieren plot of the solution looks as follows:
+
+<img src="https://www.dealii.org/images/steps/developer/step-69.coarse.gif" alt="" height="300">
+
+It is evident that 37k gridpoints for the first-order method is nowhere
+near the resolution needed to resolve any flow features. For comparison,
+here is a "reference" computation with a second-order method and about 9.5M
+gridpoints:
+
+<img src="https://www.dealii.org/images/steps/developer/step-69.2nd-order.t400.jpg" alt="" height="300">
+
+So, we give the first-order method a second chance and run it with about
+2.4M gridpoints on a small compute server:
+
+@verbatim
+$ mpirun -np 16 ./step-69 | tee output
+
+[...]
+
+ ####################################################
+ ######### #########
+ ######### Cycle 070216 (100.0%) #########
+ ######### at time t = 3.99999231 #########
+ ######### #########
+ ####################################################
+
+TimeLoop<dim>::output(t = 4.00006, checkpoint = 1)
+
++------------------------------------------------------------------------+------------+------------+
+| Total CPU time elapsed since start | 3.34e+05s | |
+| | | |
+| Section | no. calls | CPU time | % of total |
++------------------------------------------------------------+-----------+------------+------------+
+| discretization - setup | 1 | 1.96s | 0% |
+| offline_data - assemble lumped mass matrix, and c_ij | 1 | 10.6s | 0% |
+| offline_data - compute |c_ij|, and n_ij | 1 | 0.208s | 0% |
+| offline_data - create sparsity pattern and set up matrices | 1 | 0.38s | 0% |
+| offline_data - distribute dofs | 1 | 1.36s | 0% |
+| offline_data - fix slip boundary c_ij | 1 | 2.52s | 0% |
+| schlieren_postprocessor - compute schlieren plot | 201 | 24.3s | 0% |
+| schlieren_postprocessor - prepare scratch space | 1 | 0.008s | 0% |
+| time_loop - setup scratch space | 1 | 1.48s | 0% |
+| time_loop - stalled output | 200 | 0.004s | 0% |
+| time_step - 1 compute d_ij | 70216 | 5.79e+04s | 17% |
+| time_step - 2 compute d_ii, and tau_max | 70216 | 5.78e+03s | 1.7% |
+| time_step - 3 perform update | 70216 | 2.51e+04s | 7.5% |
+| time_step - 4 fix boundary states | 70216 | 56.8s | 0% |
+| time_step - prepare scratch space | 1 | 0.028s | 0% |
++------------------------------------------------------------+-----------+------------+------------+
+@endverbatim
+
+And with the following result:
+
+<img src="https://www.dealii.org/images/steps/developer/step-69.fine.gif" alt="" height="300">
--- /dev/null
+# Listing of Parameters
+# ---------------------
+subsection A - TimeLoop
+ # Base name for all output files
+ set basename = test
+
+ # Final time
+ set final time = 4
+
+ # time interval for output
+ set output granularity = 0.02
+
+ # Resume an interrupted computation.
+ set resume = false
+end
+
+
+subsection B - Discretization
+ # Immersed disc: height of computational domain
+ set immersed disc - height = 2
+
+ # Immersed disc: length of computational domain
+ set immersed disc - length = 4
+
+ # Immersed disc: diameter of immersed disc
+ set immersed disc - object diameter = 0.5
+
+ # Immersed disc: x position of immersed disc center point
+ set immersed disc - object position = 0.6
+
+ # Initial refinement of the geometry
+ set initial refinement = 5
+end
+
+
+subsection C - OfflineData
+end
+
+
+subsection D - InitialValues
+ # Initial 1d state (rho, u, p) of the uniform flow field
+ set initial 1d state = 1.4, 3, 1
+
+ # Initial direction of the uniform flow field
+ set initial direction = 1, 0
+end
+
+
+subsection E - TimeStep
+ # relative CFL constant used for update
+ set cfl update = 0.8
+end
+
+
+subsection F - SchlierenPostprocessor
+ # Beta factor used in Schlieren-type postprocessor
+ set schlieren beta = 10
+
+ # Use the corresponding component of the state vector for the schlieren
+ # plot
+ set schlieren index = 0
+end
+
+
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