Flux-Based Numerical Methods
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.
