4 | R. Altmann und C. Zimmer
**Time discretization schemes for hyperbolic systems on networks by $\varepsilon$-expansion**
*ArXiv e-print 1810.04278*, 2018. [abstract] [bibtex]
We consider partial differential equations on networks with a small parameter $\varepsilon$, which are hyperbolic for $\varepsilon>0$ and parabolic for $\varepsilon=0$. With a combination of an $\varepsilon$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $\varepsilon$-expansion. @ARTICLE{AltZ18c, archiveprefix = {arXiv}, eprint = {1807.04278}, journal = {ArXiv e-print 1810.04278}, month = {October}, pubdate = {2018-10-09}, title = {Time discretization schemes for hyperbolic systems on networks by $\varepsilon$-expansion}, url = {https://arxiv.org/abs/1810.04278}, year = {2018}, author = {Altmann, R. and Zimmer, C.}, } |

3 | R. Altmann und C. Zimmer
**On the smoothing property of linear delay partial differential equations**
*J. Math. Anal. Appl.*, Vol. 2, 2018, pp. 916–934. [abstract] [preprint] [bibtex]
We consider linear partial differential equations with an additional delay term, which - under spatial discretization - lead to ordinary differential equations with fixed delay of retarded type. This means that the semi-discrete solution gains smoothness over time. For the concept of classical, mild, and weak solutions we analyse whether this effect also takes place in the original system. We show that some systems behave in a neutral way only. As a result, the smoothness of the exact solution remains unchanged instead of gaining smoothness over time. @ARTICLE{AltZ18b, fjournal = {Journal of Mathematical Analysis and Applications}, journal = {J. Math. Anal. Appl.}, number = {467}, pages = {916--934}, preprint = {https://www3.math.tu-berlin.de/cgi-bin/IfM/show_abstract.cgi?Report-07-2017.rdf.html}, pubdate = {2018-07-26}, title = {On the smoothing property of linear delay partial differential equations}, url = {https://www.sciencedirect.com/science/article/pii/S0022247X18306310}, volume = {2}, year = {2018}, author = {Altmann, R. and Zimmer, C.}, } |

2 | A. Moses Badlyan und C. Zimmer
**Operator-GENERIC formulation of Thermodynamics of Irreversible Processes**
*ArXiv e-print 1807.09822*, 2018. [abstract] [bibtex]
Metriplectic systems are state space formulations that have become well-known under the acronym GENERIC. In this work we present a GENERIC based state space formulation in an operator setting that encodes a weak-formulation of the field equations describing the dynamics of a homogeneous mixture of compressible heat-conducting Newtonian fluids consisting of reactive constituents. We discuss the mathematical model of the fluid mixture formulated in the framework of continuum thermodynamics. The fluid mixture is considered an open thermodynamic system that moves free of external body forces. As closure relations we use the linear constitutive equations of the phenomenological theory known as Thermodynamics of Irreversible Processes (TIP). The phenomenological coefficients of these linear constitutive equations satisfy the Onsager-Casimir reciprocal relations. We present the state space representation of the fluid mixture, formulated in the extended GENERIC framework for open systems, specified by a symmetric, mixture related dissipation bracket and a mixture related Poisson-bracket for which we prove the Jacobi-identity. @ARTICLE{MosZ18, archiveprefix = {arXiv}, eprint = {1807.09822}, journal = {ArXiv e-print 1807.09822}, month = {July}, pubdate = {2018-07-25}, title = {Operator-{GENERIC} formulation of Thermodynamics of Irreversible Processes}, url = {https://arxiv.org/abs/1807.09822}, year = {2018}, author = {{Moses Badlyan}, A. and Zimmer, C.}, } |

1 | R. Altmann und C. Zimmer
**Runge-Kutta methods for linear semi-explicit operator differential-algebraic equations**
*Math. Comp.*, Vol. 87, 2018, pp. 149–174. [abstract] [preprint] [bibtex]
As a first step towards time-stepping schemes for constrained PDE systems, this paper presents convergence results for the temporal discretization of operator DAEs. We consider linear, semi-explicit systems which includes e.g. the Stokes equations or applications with boundary control. To guarantee unique approximations, we restrict the analysis to algebraically stable Runge-Kutta methods for which the stability functions satisfy $R(\infty)=0$. As expected from the theory of DAEs, the convergence properties of the single variables differ and depend strongly on the assumed smoothness of the data. @ARTICLE{AltZ18a, title = {{R}unge-{K}utta methods for linear semi-explicit operator differential-algebraic equations}, journal = {Math. Comp.}, fjournal = {Mathematics of Computation}, year = {2018}, volume = {87}, number = {309}, pages = {149--174}, doi = {10.1090/mcom/3270}, url = {http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2017-03270-1/}, pubdate = {2017-06-21}, preprint = {http://www3.math.tu-berlin.de/cgi-bin/IfM/show_abstract.cgi?Report-10-2016.rdf.html}, author = {Altmann, R. and Zimmer, C.}, } |