5.3.1 Neumann Type (Explicit Flux)

In case an expression for the flux F_{$\nu_{i,i'}$} is available, the constitutive relations can easily be completed by

$$\nu_{i} \nu_{i} \nu_{i,i'}$$ = 0 (5.9)

$$\nu_{i'} \nu_{i'} \nu_{i,i'}$$ = 0 (5.10)

To accomplish this the respective part of the transformation matrix reads

tx, y	$\nu_{i}^{}$	$\nu_{i'}^{}$
$\nu_{i}^{}$	1
$\nu_{i'}^{}$		1

which is actually the default case and hence needs not to be explicitly specified.

The derivatives of the flux term with respect to the solution variables must be of the same magnitude as the other derivatives given by f^S_{$\nu_{i}$} in order to guarantee good conditioning of the resulting equation system [15]. Since this cannot generally be guaranteed, an approach similar to Dirichlet interfaces is preferable [63]: Adding (5.9) to (5.10) one gets

$$\nu_{i} \nu_{i} \nu_{i,i'}$$ = 0 (5.11)

$$\nu_{i'} \nu_{i} \nu_{i'}$$ = 0 (5.12)

As for Dirichlet interfaces, (5.11) is pre-eliminated before the actual iterative solution process. The appropriate transformation matrix reads

tx, y	$\nu_{i}^{}$	$\nu_{i'}^{}$
$\nu_{i}^{}$	1
$\nu_{i'}^{}$	1	1