Kinetic and hyperbolic equations with applications to engineering processes
- Kinetik und Hyperbolische Gleichungen mit Anwendungen in ingenieurwissenschaftlichen Prozessen
Häck, Axel-Stefan; Herty, Michael Matthias (Thesis advisor); Banda, Mapundi K. (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2017
The content of this doctoral thesis is structured in six chapters. Chapters 1-3 are devoted to introduce widely known and well established concepts and results that are of use to understand Chapters 4-5. In the latter chapters original research by — among others — the author is presented. The last Chapter 6 is the obligatorily ”Summary and Conclusion”.In Chapters 1 we undergo the derivation of the Botlzmann equation for a gas from the underlying particle dynamics of single atoms. After establishing the Botlzmann equation we introduce moments of the probability density and additionally take a hydrodynamical limit to derive the Euler equations for gas dynamics.The second chapter is devoted to hyperbolic conservation laws — in the spatial one-dimensional case. We introduce popular cases of such conservation laws. Then we discuss the Riemann problem. After establishing a theory to solve a general Riemann problem we use this concept to introduce the wave front tracking algorithm. In the last section of this chapter we prove the existence of solutions to general initial value problems of hyperbolic conservation laws.In Chapter 3 we are interested in the numerical analysis of conservation laws. First we give a general description of finite volume methods. The particular finite volume methods distinguish by the different numerical fluxes. We introduce a variety of different numerical fluxes and analyze them theoretically. Following to this we present three test cases and apply those finite volume schemes on them. After this we establish spacial two-dimensional finite volume methods. We apply those two-dimensional schemes to a set of Riemann problems and compare their performance.In the fourth Chapter we are interested in the mathematical modeling of steel rolling processes. Our basic tool will be kinetic partial differential equations suitable for large time scales and for many roll passes, similarly to the discussion in the first chapter. We start with an overview of the rolling process and typical process models of it. Analog to the fist chapter, we then define a particle based model describing a steel rolling process — in a simplified manner. After this we will derive a kinetic equation associated to this process and finally perform a hydrodynamic limit to this kinetic model in order to attain a fluid dynamical model. We then give numerical solutions to this fluid-like model, using the (two-dimensional) schemes discussed in Chapter 3.In Chapter 5 we develop a second order finite volume scheme for general 2 × 2 hyperbolic systems on networks. The crucial point is the derivation of a suitable numerical flux at the nodal point at the juncture where multiple arcs are connected. We use a characteristic decomposition of the temporal derivative up to second order of the solution at the nodal point and estimate the outgoing information using spatial derivatives. After established the scheme theoretically we then present numerical results for gas flow in pipe networks, using a hyperbolic conservation law introduced in Chapter 3.As already mentioned is the final chapter devoted to give a brief summary and a conclusion of the new results established in Chapter 4 and Chapter 5.