A Unified Framework for Element-Based High-Order Summation-By-Parts Operators on Unstructured Grids

• Applicant: Professor Dr.-Ing. Gregor Gassner
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 266390124

This project deals with the construction and analysis of a novel mathematical and computational framework, which unifies several different high order methods for the discretisation of non-linear advection-diffusion problems. The general formulation includes the class of nodal discontinuous Galerkin (DG), finite volume (FV), finite difference (FD) and finite element (FE) methods as long as the operators satisfy the so-called summation-by-parts property (SBP). SBP is a discrete analogon to integration-by-parts and is an essential mathematical ingredient to show stability and exact conservation of a high order discretization used to approximate conservation laws. As long as a discretization satisfies SBP, it can be formulated in a common framework and can be implemented into the same computational framework. The general frame of this formulation is a DG scheme, where the approximation "inside" the grid cell is exchanged by other discretisations such as e.g. a FD scheme. This enables a direct comparison of these methods with respect to accuracy and computational efficiency and hence allows us to classify all these methods, which is an important quest in the "high order community". Furthermore, the DG nature of the discretisations already supports parallel computing and hence the resulting computational framework enables efficient use of the largest super computers. Another main objective of this project is to construct a stable and accurate interface procedure, which allows to chose in every grid cell a different discretisation method. This allows us to run FD, FV, FE and DG methods side by side in the same computation. All of this is based on a unified mathematical formulation and consequently on a common computational implementation. This unified framework opens up a multitude of new possibilities, some of which are directly explored in the second part of the proposed project and others are intended for a future second phase of this project, such as e.g. multi-physics simulations. The first idea is that this unified framework enables operator adaptation in the sense that we can adapt the discretisation cell by cell according to the underlying structure of the solution. As a complex application, the aeroacoustics of a turbulent flow past a wing is considered. Using the results from the assessment of the different discretisations in the first part of the project, we choose optimal operators to capture the boundary layer, the vortex street, and the far-field propagation of the acoustic waves. Furthermore, we use FD approximations at the boundary of the domain and use the vast work on non-reflecting artificial boundary conditions available for such discretisations. This is a second application of this unified framework since a pure DG discretization would be problematic at the boundaries as up to now, no suitable non-reflecting boundary treatment is known for DG methods.

• DFG Programme: Research Grants