Ein einheitliches Framework für Element basierte Summation-by-Parts Operatoren hoher Ordnung auf unstrukturierten Gittern
Zusammenfassung der Projektergebnisse
The major scientific outcome of the project is a fundamental deep understanding of how to construct non-conforming approximations that are conservative, linear and/or non-linear stable while retaining accuracy for the simulation of hyperbolic conservation laws. Hyperbolic conservation laws form an important class of (possible non-linear) partial differential equations, used to model for instance gas dynamics (Euler equations), electrodynamics (Maxwell’s equations), tsunami propagation (shallow water equations), and e.g. plasma physics (magnetohydrodynamic equations). Solutions of these problems are typically very complex with multiple scales in space (and time) and even discontinuities such as shocks. It is thus highly challenging for a numerical method to solve these problems in an efficient way. One silver bullet to tame the beast by drastically increasing the efficiency of the simulation is the idea of adaptivity. Adaptivity in the sense that the approximative scheme controls its accuracy according to the local solution features and consequently locally adapts its approximation space. Adaptivity typically results in so-called non-conforming approximation spaces, where grid nodes at internal domain interfaces do not match. It is thus an important question how to glue these nonmatching (or non-conforming) internal interfaces together. For hyperbolic conservation laws, several mathematical criteria are important for a valid numerical method, such as e.g.: (i) conservation of the primary quantities such as mass, momentum and energy and (ii) entropy stability, i.e. satisfaction of the second law of thermodynamics. It is important to note that these properties directly depend on how the non-conforming interfaces are glued together. The outcome of the project is fundamental and deep insight in the necessary discrete conditions for a numerical approximation, such that the important mathematical criteria (i) and (ii) are satisfied. In addition, a precise guideline how to construct discrete operators that satisfy the conditions. The keys are (i) specific projection operators operators that connect the non-conforming interfaces, and (ii) construction of specific (non-linear) fluxes that directly depend on these projection operators. With this, we are able to present the first provably entropy stable framework for arbitrary non-conforming approximations of non-linear hyperbolic conservation laws. We emphasise that the mathematical proofs are general for all summation-by-parts operators with diagonal mass matrix, including Legendre- Gauss-Lobatto nodal discontinuous Galerkin methods and finite difference methods. Furthermore, the proofs are general for arbitrary non-conforming interfaces (node distribution, arbitrary number of hanging nodes), and for all non-linear hyperbolic conservation laws, where a discrete conforming entropy analysis exists.
Projektbezogene Publikationen (Auswahl)
- An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property. Journal of Scientific Computing
Lucas Friedrich, Andrew R. Winters, David C. Del Rey Fernandez, Gregor J. Gassner, Matteo Parsani, Mark H. Carpenter
- Conservative and Stable Degree Preserving SBP Operators for Non-Conforming Meshes. Journal of Scientific Computing 75(2): 657-686 (2018)
L. Friedrichs, D. C. D. R. Fernandez, A. R. Winters, G. J. Gassner, D. W. Zingg, and J. Hicken
(Siehe online unter https://doi.org/10.1007/s10915-017-0563-z)
