Summation-by-parts formulations for flow problems

Sammanfattning: Many problem in engineering and physics can be described by partial differential equations (PDEs). Augmented with proper initial and boundary conditions, the PDE forms an initial-boundary value problem (IBVP), which is said to be well-posed if a unique solution exists that is bounded by the given data. The behavior of the solution as well as the well-posedness of the IBVP are heavily influenced by the form and position of the boundary conditions. Although it exists, the unique analytic solution to a general IBVP cannot be obtained in closed form and we must instead aim for its approximation using numerical methods. Summation-by-parts (SBP) operators together with the Simultaneous approximation term (SAT) technique can be used to form stable numerical methods that generate accurate approximations. All discretizations in this thesis are based on the SBP-SAT framework.In computational fluid dynamics, the numerical domain is often truncated and data must be imposed at artificial boundaries. The first and major part of this thesis focuses on how different formulations of boundary conditions commonly applied at open boundaries affect the simulations. Stability and spectral properties of the solution are investigated using the energy method and normal mode analysis. Once the continuous problem is analyzed, we proceed to the SBP-SAT formulation, which allows us to discretely mimic the continuous analysis, and verify the theoretical findings.Close to solid walls, fluids may develop steep gradients that require a fine mesh for an accurate simulation of the turbulent boundary layer. One strategy to resolve such flows is to use a wall model instead of a fine mesh. However, introducing a wall model may lead to instabilities. In the second part of the thesis, we highlight this issue and propose new energy-stable boundary procedures that are accurate even on coarse meshes.Stability is key for a numerical scheme but in some applications not enough, and other properties are desired by the discretization. In the third and last part of this thesis, we derive special SBP operators for an application in high-energy physics that preserves the trace of the so-called density matrix, a quantity that is also preserved in the continuous formulation.

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