Guideline

mpirun -np 1 ./TauBench -n 250000 -s 10

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 defined as

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 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 (Option 2b)

 

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)