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- Laurén, Fredrik, 1990-
(author)
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Summation-by-parts formulations for flow problems
- 2022
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Doctoral thesis (other academic/artistic)abstract
- Many problems 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). An IBVP is said to be well-posed if a unique solution bounded by the given data exists. 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. The solution, if it exists, to an IBVP cannot in general be obtained in closed form. Instead, one can compute an 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.The first part of the thesis concentrates on IBVPs describing fluid motion. Different sets of boundary conditions are investigated in terms of energy-boundedness and spectral properties. Wall models are also studied and used to aid coarse-grid simulations in an energy stable manner.In the second part of the thesis, special discretization operators are derived. First, single-block SBP operators are combined with interpolation operators. The result is a single SBP operators on a multi-block grid that encapsulates both the metric terms and the interface treatments. Second, new SBP operators are developed for an application in high-energy physics. The new operators preserves the unit-trace, which is important since it enables a probability interpretation of the density matrix.
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| 2. |
- Ålund, Oskar, 1987-
(author)
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Applications of summation-by-parts operators
- 2020
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Doctoral thesis (other academic/artistic)abstract
- Numerical solvers of initial boundary value problems will exhibit instabilities and loss of accuracy unless carefully designed. The key property that leads to convergence is stability, which this thesis primarily deals with. By employing discrete differential operators satisfying a so called summation-by-parts property, it is possible to prove stability in a systematic manner by mimicking the continuous analysis if the energy has a bound. The articles included in the thesis all aim to solve the problem of ensuring stability of a numerical scheme in some context. This includes a domain decomposition procedure, a non-conforming grid coupling procedure, an application in high energy physics, and two methods at the intersection of machine learning and summation-by-parts theory.
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