# Entropy methods for gas dynamics on networks

• Entropie Methoden zur Gasdynamik auf Netzwerken

Aachen (2020, 2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

In the last two decades, the simulation of gas dynamics on networks became of huge interest. This can be traced back to the various applications as well as the interesting mathematical structures and problems. The network models are based on one-dimensional hyperbolic conservation laws coupled at the junctions in a suitable way. They were studied in various aspects (e.g. analysis, numerics, optimization or control theory) and different coupling techniques were introduced. Besides gas dynamics there are various applications of network models, e.g. traffic flow, supply chains, data networks, blood flow or water channels. We follow an approach based on a kinetic model for isentropic gas dynamics. Therefore, we introduce kinetic coupling conditions and solve the kinetic model along the edges. We prove existence of finite mass and energy solutions under weak assumptions on the initial data and coupling conditions. We justify the relaxation of the kinetic solutions towards a macroscopic solution to the system of isentropic gas dynamics by using the method of compensated compactness. The obtained solutions satisfy inherited entropy flux inequalities at the junction and the method applies to a large class of kinetic coupling conditions. As a byproduct, we get an existence result for solid wall boundary conditions and generalize the approach to networks with finitely many junctions. An entropy principle leads to the kinetic coupling condition which dissipates as much entropy as possible under the constraint of conserved mass. The outgoing characteristics are given by the Maxwellian corresponding to a macroscopic state with suitable artificial density and zero speed. The kinetic condition leads to a macroscopic condition based on standard Riemann problems with a left initial state given by a suitable artificial density and zero speed. We call this condition the artificial density coupling condition. For the artificial density coupling condition, existence and uniqueness of solutions to the generalized Riemann problem will be proven globally in state space. Existence and uniqueness of solutions to the generalized Cauchy problem will be shown under the standard assumption of sufficiently small total variation of the initial data. The obtained solutions satisfy entropy flux inequalities at the junction which imply non-increasing energy at the junction and a maximum principle on the Riemann invariants. Numerical simulations demonstrate that the artificial density coupling condition is the only known condition leading to the physically correct wave types in some examples. The new approach generalizes to full gas dynamics and leads to a coupling condition based on an artificial state with suitable density, energy and zero speed with non-increasing entropy at the junction. We also present new results regarding boundary conditions for hyperbolic conservation laws. Boundary conditions are strongly related to the artificial density coupling condition since the coupling condition can be understood as boundary conditions with boundary data given by the artificial density and zero speed. We give new conditions under which the entropy and Riemann problem formulation of boundary conditions are equivalent or not equivalent. In the case of isentropic gas dynamics, we use numerical methods and Kružkov-type entropies to study the differences between both formulations globally in state space.