### Flux-Based Numerical Methods

The methodology of this research unit is build upon the fact that the partial differential equations
for all descriptions of all the processes can be written in conservation or balance law form with a well-defined
flux function. The most promising way to provide structure-preserving numerical methods
for bulk- and interface-coupling focuses on numerical fluxes. The most prominent example is
the conservative discretization of the finite volume method.

In fact, the interface-coupling to another PDE model is then realized only through the interface
via the numerical fluxes. The scientific challenge lies in the construction and the definition of such
numerical fluxes. If we assume that the local discretization preserves the desired structure of the
PDE, which is already difficult in its own, the question is: Is it possible to define numerical fluxes
at the interface of the coupling boundary such that the structure is still preserved across the PDE
models independently of the discretization parameter h?
The local data dependence is also computationally beneficial when considering bulk-coupling of
different PDE models. Again, the additional challenge lies in the goal to construct numerical approximations
that not only preserve the structures of a single PDE models, but preserve structures
across several PDE models and their discretizations. Coupling of several PDE models certainly
means exchange of data through the volume. Since the amount of exchanged data is large compared
to interface-coupling, it is important to partition the computational domain such that coupling
data transferred via network communication is minimized. This can be achieved by assigning cells of
coupled systems that need to exchange data to the same subdomain. For a local compact operator,
this exchange can then be done in local memory only. Numerical approximations with large data
stencils and dependencies, however, might need to transfer a significant share of the coupling data,
and which are hence not available in local memory. Compact flux-oriented discretizations
have the additional benefit of being very well-suited for domain decomposition-based parallelization,
as no large stencil data needs to be communicated via the network, but only boundary data.

Structure Preservation
Code Development