Czernik, Christof (2022). Entropy Stable Discontinuous Galerkin Methods for Multi-Component Euler and Ideal Magnetohydrodynamics Equations with Chemical Networks in Julia. PhD thesis, Universität zu Köln.

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We present a high-order entropy-stable discontinuous Galerkin spectral element method (DGSEM) for multi-component Euler and multi-component ideal magnetohydrodynamics (MHD) equations with chemical reaction terms written in Julia. Instead of using a completely self-made code, we extend the already existing and proven simulation framework Trixi.jl, so that we can make our new introduced features available to the public. For this purpose, we extend the simulation framework Trixi.jl with multi-component Euler and multi-component ideal MHD equations. Since we place value on entropy-stable processes, we add an entropy-conservative flux function for the multi-component Euler equations from the literature and propose an entropy-conservative flux function for the multi-component ideal MHD equations. Trixi.jl contains a very effective shock-capturing method where the high-order DGSEM can be blended with a first-order Finite Volume (FV) scheme for cartesian meshes. To be able to simulate applications with more complex geometries we are going to extend this feature to unstructured and curvilinear meshes and make it work for multi-component equations. Another feature in Trixi.jl is the positivity-preserving limiter, which is able to rescue a solution in difficult situations. In the literature, however, another method has emerged which, based on the shock-capturing method used in Trixi.jl, is able to preserve the positivity of density and pressure for single-component simulations. In this work we add this new positivity-preserving scheme to Trixi.jl and propose slight modifications for the multi-component case. The new multi-component equations give us the opportunity to introduce chemical reactions into Trixi.jl. Since we advocate the use of packages in this work, we will add an external package specialized on the solution of chemical reactions and give an overview and introduction to all important packages in Trixi.jl. Finally, we provide numerical test cases that verify the theoretical properties of the new introduced features and demonstrate the strengths and weaknesses of our method. Additionally, we demonstrate the capabilities of our method with complex numerical examples.

Item Type: Thesis (PhD thesis)
CreatorsEmailORCIDORCID Put Code
Czernik, Christofcczerni1@uni-koeln.deUNSPECIFIEDUNSPECIFIED
URN: urn:nbn:de:hbz:38-643276
Date: 18 September 2022
Language: English
Faculty: Faculty of Mathematics and Natural Sciences
Divisions: Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects: Mathematics
Uncontrolled Keywords:
Entropy StableEnglish
Euler EquationsUNSPECIFIED
Date of oral exam: 28 November 2022
NameAcademic Title
Gassner, GregorProf. Dr.
Birken, PhilippProf. Dr.
Refereed: Yes


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