- All presenters in the workshop should
complete at least one sub-case from one of the 13 test cases.
- For C1 cases, hp-adaptations are required
to obtain an accurate ¡°exact¡± solution for error computation unless
entropy errors are used as the indicator. For other cases, hp-refinement
studies should be performed with at least three data points to demonstrate
- The cost of the computation should be
expressed in work units. TauBench (here)
should be run at least three times to obtain an average wall clock time T1.
Then track the wall clock time taken by your CFD solver (excluding the
initialization, post-processing data preparation time and file I/O time) T2.
The work unit is then defined as T2/T1. When running
TauBench, use the following parameters:
mpirun ¨Cnp 1 ./TauBench ¨Cn 250000
- The length scale h in all computations will be defined as for 2D
problems, and for 3D
problems, with the
total number of degrees of freedom per equation, unless otherwise
- For steady problems, start your
computation from a uniform free-stream unless otherwise specified. Use the
L2 norm of the density residual to monitor convergence. Steady
state is assumed if the initial residual is dropped by 10 orders of
magnitude. For cases impossible to converge 10
orders, 8 orders can be used as a convergence criterion. For the flat
plate boundary layer problem, use the L2 norm of the x-momentum
residual to monitor convergence.
- For each p (order of accuracy), compute the work units for 100 residual
evaluations with 250,000 degrees-of-freedom per equation. Use your finest
mesh, and scale your results for 250,000 DOFs. Submit the results to the
workshop e-mail address.
If you compute more than one case, submit your results using separate
messages. Put ¡°Case CX.X Results¡± in your subject line. Submit all results
The gmsh format (http://www.geuz.org/gmsh/doc/texinfo/gmsh.html)
is adopted for the workshop. The latest user¡¯s
manual is also here. Computational meshes in the
same refinement sequence are solicited from the participants (but not
required to participate). Good meshes will be posted on the web site to
serve as common meshes for all participants.
definition for convergence monitoring: It is not trivial to define a residual
easily computable for all methods. Consider
The integration of the equation on element Vi is
Now replacing the normal flux term with any Riemann flux as the
numerical flux, we obtain
where is the reconstructed
approximate solution on Vi,
and is the solution outside Vi. The element residual is
Then the L2
norm of the residual is defined as
where N is the total number of elements or cells. For a node based finite
difference method, it is ok to use as the residual definition. These two
definitions are expected to differ by a second order term.
Furthermore, note that the definition of Res_i above is an example only. For
different equations and different discretizations considered the definition of
Res_i needs to be modified to coincide with the discretization-specific
residual of the scheme.
For any solution variable (preferably non-dimensional) s, the L2 error is defined as (Option 1)
For an element or cell based
method (FV, DG etc), where a solution distribution is available on the element,
the element integral should be computed with a quadrature formula of sufficient
precision, such that the error is nearly independent (with 3 significant
digits) of the quadrature rule. Note that for a FV method, the reconstructed
solution should be the same as that used in the actual residual evaluation.
For a finite difference
scheme, if the Jacobian matrix is available, i.e.,
the L2 error is defined as (Option 2a)
Otherwise, the L2 error is defined as
For some numerical methods, an error defined
based on the cell-averaged solution may reveal super-convergence properties. In
such cases, we suggest another definition (Option 3a)
In this definition, one can also drop the volume
in a similar fashion to the definition for finite-difference type methods,
i.e., (Option 3b)