Analysis and numerical methods for coupled hyperbolic conservation laws
Sikstel, Aleksey; Herty, Michael (Thesis advisor); Müller, Siegfried (Thesis advisor); Rohde, Christian (Thesis advisor)
Dissertation / PhD Thesis
This thesis is devoted to the study of analytic properties of solutions to coupled conservation laws and associated numerical methods. The focus is put on two particular problems: (i) a compressible gas coupled by means of a gas generator and (ii) a linear elastic material coupled with a compressible fluid. The analytical and numerical methods developed here are applicable to a broad class of coupled problems involving hyperbolic conservation laws. In Chapter 2 systems of hyperbolic conservation laws are defined and basic concepts, namely, weak solutions, Rankine-Hugoniot conditions, entropy solutions and Laxcurves are introduced. Based on these concepts, local solutions to the Riemann problem for general one-dimensional systems of hyperbolic conservation laws are presented. In Chapter 3, the Riemann problem for the linear elastic equations and the Euler equations with perfect gas equation of state are solved for arbitrary physically admissible initial conditions. Although the topics considered so far are classical and wellknown, they are required in the subsequent chapters. In Chapter 4 the general framework for the coupling of systems of hyperbolic conservation laws by means of Riemann problems is introduced. This framework is subsequently applied to the fluid-structure coupling problem and to Euler equations coupled by a gas generator. The conditions for the existence of unique solutions of each coupling problem is investigated. Chapter 5 deals with numerical methods for coupling problems. A procedure for the realisation of the general coupling framework from Chapter 4 incontext of Runge-Kutta discontinuous Galerkin methods is presented. Moreover, a multilevel time stepping algorithm for coupled problems with explicit time stepping schemesis introduced. Numerical results of several simulations for both the fluid-structure and the gas gen-erator coupling problem are presented and investigated. Three test cases for the gas generator coupling problem are considered. Two of the cases are supposed to model a realistic situation while the third one postulates extreme conditions that do not occur in the real world, however, are useful to test the proposed methods. Moreover, the empirical order of convergence of the proposed numerical coupling method is validated for a fluid structure coupling solution of a shock hitting the solid-fluid interface, transmitted into the solid part and reflected into the fluid part. In addition, the empirical order of convergence of the multilevel time stepping method is computed for a similar test case. The result indicates that the multilevel time stepping method does not spoil the accuracy while saving a considerable amount of superfluous time steps in the fluid part. Finally, a test case of a two-dimensional hot bubble collapsing in the surrounding cold fluid is presented. The fluid part is modelled by the compressible Euler equations with stiﬀened gas equation of state and the multilevel time stepping method is applied. The solution to this test case exhibits a complex wave structure arising from waves that are emitted from the collapsing bubble. These waves are transmitted and reflected at the solid-fluid interface and subsequently interact with the hot bubble. Chapter 6 concludes the thesis and gives a brief outlook.